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Fq(of)g(matrices)996 3096 y Fp(\033)1048 3111 y Fk(a;b)1179 3096 y Fq(=)1291 2968 y Fj(\022)1373 3040 y Fp(a)94 b(b)1374 3153 y Fq(0)f(1)1572 2968 y Fj(\023)1836 3096 y Fp(a;)15 b(b)26 b Fn(2)f Fp(K)q(;)107 b(a)25 b Fn(6)p Fq(=)g(0)-24 3345 y(whic)m(h)k(w)m(e)i(let)f(act)i(on)e (irreducible)d(p)s(olynomials)g(as)k(follo)m(ws)1200 3550 y Fp(\033)s Fq(\()p Fp(f)10 b Fq(\)\()p Fp(x)p Fq(\))26 b(=)f Fp(a)1672 3512 y Fi(\000)p Fl(deg)r Fk(f)1879 3550 y Fp(f)10 b Fq(\()p Fp(a)15 b(x)20 b Fq(+)g Fp(b)p Fq(\))p Fp(:)1064 b Fq(\(11\))-24 3754 y(The)34 b(p)s(olynomial)d Fp(\033)s Fq(\()p Fp(f)10 b Fq(\))35 b(is)e(monic)h(and)f(irreducible,) f(and)i(since)g Fp(\033)2342 3769 y Fk(c;d)2432 3754 y Fq(\()p Fp(\033)2519 3769 y Fk(a;b)2611 3754 y Fq(\()p Fp(f)10 b Fq(\)\))32 b(=)g(\()p Fp(\033)2993 3769 y Fk(c;d)3084 3754 y Fp(\033)3136 3769 y Fk(a;b)3227 3754 y Fq(\)\()p Fp(f)10 b Fq(\),)36 b Fp(G)-24 3867 y Fq(de\014nes)28 b(an)h(action)h(on)f Fp(E)5 b Fq(\()p Fp(f)10 b Fq(\),)30 b(or)f(on)g(its)g(graph.)39 b(The)29 b(orbits)f(of)h Fp(G)g Fq(partition)f(the)h(v)m(ertices)h(of)f(ev)m(ery)-24 3980 y(extension)h(graph)g(in)m(to)g Fo(clusters.)117 4093 y Fq(W)-8 b(e)32 b(consider)d(the)h(additiv)m(e)g(normal)f (subgroup)795 4342 y Fp(G)867 4304 y Fl(+)951 4342 y Fq(=)24 b Fn(f)p Fp(\033)1143 4357 y Fl(1)p Fk(;b)1233 4342 y Fn(g)333 b Fp(\033)1663 4357 y Fl(1)p Fk(;b)1792 4342 y Fq(=)1903 4214 y Fj(\022)1985 4285 y Fq(1)95 b Fp(b)1985 4398 y Fq(0)d(1)2182 4214 y Fj(\023)2446 4342 y Fp(b)26 b Fn(2)e Fp(K)q(:)659 b Fq(\(12\))-24 4591 y(Its)35 b(orbits)g(are)h Fo(blo)-5 b(cks,)37 b Fq(whic)m(h)e(sub)s (divide)d(eac)m(h)k(cluster.)56 b(Lik)m(ewise,)37 b(w)m(e)f(ha)m(v)m(e) g(the)g(m)m(ultiplicativ)m(e)-24 4704 y(group)708 4862 y Fp(G)780 4824 y Fi(\003)844 4862 y Fq(=)25 b Fn(f)p Fp(\033)1037 4876 y Fk(a;)p Fl(0)1134 4862 y Fn(g)333 b Fp(\033)1564 4876 y Fk(a;)p Fl(0)1701 4862 y Fq(=)1812 4733 y Fj(\022)1894 4805 y Fp(a)91 b Fq(0)1895 4918 y(0)i(1)2093 4733 y Fj(\023)2357 4862 y Fq(0)26 b Fn(6)p Fq(=)f Fp(a)g Fn(2)g Fp(K)q(:)572 b Fq(\(13\))1725 5225 y(7)p eop %%Page: 8 8 8 7 bop -24 573 a Fq(W)-8 b(e)31 b(in)m(tro)s(duce)f(the)g(short-hand)g (notation)1033 777 y Fp(f)1088 739 y Fl(+)1078 806 y Fk(b)1172 777 y Fq(:=)25 b Fp(\033)1345 792 y Fl(1)p Fk(;b)1434 777 y Fq(\()p Fp(f)10 b Fq(\))364 b Fp(f)1978 740 y Fi(\003)1968 800 y Fk(a)2042 777 y Fq(:=)25 b Fp(\033)2215 791 y Fk(a;)p Fl(0)2312 777 y Fq(\()p Fp(f)10 b Fq(\))p Fp(;)896 b Fq(\(14\))-24 981 y(and)35 b(w)m(e)i(denote)f(the)g(blo)s(c) m(k)g(con)m(taining)f Fp(f)45 b Fq(b)m(y)36 b(\002)1744 996 y Fk(f)1789 981 y Fq(,)i(so)e(that)h(\002)2243 996 y Fk(f)2337 981 y Fq(=)50 b Fn(f)p Fp(f)2558 943 y Fl(+)2548 1011 y Fk(b)2632 981 y Fn(j)15 b Fp(b)35 b Fn(2)f Fp(K)7 b Fn(g)p Fp(:)36 b Fq(If)g(\002)3199 995 y Fk(g)3273 981 y Fq(=)e(\002)3449 996 y Fk(h)3494 981 y Fq(,)-24 1094 y(then)c Fp(h)d Fq(=)e Fp(g)404 1056 y Fl(+)401 1124 y Fk(b)464 1094 y Fq(,)31 b(for)f(some)i Fp(b)p Fq(.)41 b(But)31 b(then)g Fp(h)26 b Fq(=)g Fp(f)j(.)21 b(g)35 b Fq(with)29 b Fp(f)10 b Fq(\()p Fp(x)p Fq(\))26 b(=)g Fp(x)20 b Fn(\000)g Fp(b)p Fq(,)31 b(that)h(is,)e(the)h(action)g (of)g Fp(G)3461 1061 y Fl(+)-24 1207 y Fq(can)g(b)s(e)e(represen)m(ted) i(in)e(terms)h(of)g(the)h(w)m(edge)g(op)s(erator.)117 1320 y(The)e(follo)m(wing)g(t)m(w)m(o)i(lemmas)f(collect)g (miscellaneous)e(results)h(on)h(the)g(action)g(of)g Fp(G)p Fq(,)g(to)h(b)s(e)e(used)-24 1433 y(in)g(later)h(sections.)41 b(W)-8 b(e)32 b(denote)f(the)f(discriminan)m(t)e(of)i Fp(f)40 b Fq(b)m(y)30 b(\001\()p Fp(f)10 b Fq(\).)-24 1645 y Fm(Lemma)33 b(3.1)46 b Fo(L)-5 b(et)29 b Fp(f)38 b Fo(and)30 b Fp(g)j Fo(b)-5 b(elong)29 b(to)h(a)f(stable)h(set,)g(and) f(let)h Fp(n)24 b Fq(=)h(deg)r Fp(f)10 b Fo(.)40 b(The)29 b(action)h(of)f Fp(G)g Fo(enjoys)-24 1758 y(the)k(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties)56 1946 y Fq(\()p Fp(i)p Fq(\))46 b Fo(F)-7 b(or)34 b(every)e Fp(\033)d Fn(2)c Fp(G)32 b Fo(we)g(have)h Fp(\033)s Fq(\(\002)1388 1961 y Fk(f)1434 1946 y Fq(\))26 b(=)f(\002)1662 1965 y Fk(\033)r Fl(\()p Fk(f)7 b Fl(\))1804 1946 y Fp(:)24 2134 y Fq(\()p Fp(ii)p Fq(\))47 b Fo(F)-7 b(or)34 b(every)e Fp(\033)665 2149 y Fk(a;b)782 2134 y Fn(2)25 b Fp(G)32 b Fo(we)h(have)g Fp(\033)1368 2149 y Fk(a;b)1459 2134 y Fq(\()p Fp(f)10 b Fq(\))20 b Fp(.)h(\033)1722 2149 y Fk(a;b)1813 2134 y Fq(\()p Fp(g)s Fq(\))42 b(=)e Fp(\033)2134 2148 y Fk(a)2171 2128 y Ff(n)2214 2148 y Fk(;)p Fl(0)2288 2134 y Fq(\()q Fp(f)29 b(.)21 b(g)s Fq(\))16 b Fp(:)-7 2321 y Fq(\()p Fp(iii)p Fq(\))47 b(\001\()p Fp(\033)366 2336 y Fk(a;b)458 2321 y Fq(\()p Fp(f)10 b Fq(\)\))40 b(=)h(\001\()p Fp(f)10 b Fq(\))p Fp(=a)1064 2288 y Fk(n)h Fl(\()p Fk(n)p Fi(\000)p Fl(1\))1310 2321 y Fo(.)8 2509 y Fq(\()p Fp(iv)s Fq(\))47 b Fo(If)33 b Fp(f)5 b(;)15 b(g)28 b Fn(2)d Fq(\002)p Fo(,)32 b(then,)h(for)g(al)5 b(l)33 b Fp(b)26 b Fn(2)f Fp(K)7 b Fo(,)32 b(we)g(have)h Fp(f)d(.)21 b(f)2016 2471 y Fl(+)2006 2538 y Fk(b)2099 2509 y Fq(=)k Fp(g)f(.)d(g)2374 2471 y Fl(+)2371 2538 y Fk(b)2433 2509 y Fo(.)-24 2721 y(Pr)-5 b(o)g(of.)84 b Fq(If)30 b Fp(g)f Fn(2)c Fq(\002)618 2736 y Fk(f)663 2721 y Fq(,)31 b(then)f Fp(\033)981 2688 y Fl(+)1040 2721 y Fq(\()p Fp(f)10 b Fq(\))25 b(=)g Fp(g)34 b Fq(for)c(some)h Fp(\033)1785 2688 y Fl(+)1869 2721 y Fn(2)25 b Fp(G)2027 2688 y Fl(+)2086 2721 y Fq(,)30 b(whence)1088 2926 y Fp(\033)s Fq(\()p Fp(g)s Fq(\))c(=)f Fp(\033)s(\033)1491 2888 y Fl(+)1551 2926 y Fq(\()p Fp(f)10 b Fq(\))25 b(=)g(\()p Fp(\033)s(\033)1942 2888 y Fl(+)2002 2926 y Fp(\033)2057 2888 y Fi(\000)p Fl(1)2151 2926 y Fq(\))15 b Fp(\033)s Fq(\()p Fp(f)10 b Fq(\))p Fp(:)-24 3130 y Fq(Because)39 b Fp(G)403 3097 y Fl(+)498 3130 y Fq(is)e(a)h(normal)e(subgroup,)i(w)m(e)g(ha)m(v)m(e)g(that)g Fp(\033)s(\033)2095 3097 y Fl(+)2170 3130 y Fp(\033)2225 3097 y Fi(\000)p Fl(1)2356 3130 y Fn(2)f Fp(G)2526 3097 y Fl(+)2585 3130 y Fq(,)i(so)f(that)g Fp(\033)s Fq(\()p Fp(g)s Fq(\))g(is)f(in)f(the)-24 3243 y(same)27 b(blo)s(c)m(k)f(as)i Fp(\033)s Fq(\()p Fp(f)10 b Fq(\).)39 b(F)-8 b(urthermore,)28 b Fp(\033)i Fq(is)c(injectiv)m(e,)h(b)s(ecause)g(\(11\))i(de\014nes)c (a)j(group)e(action.)40 b(This)-24 3356 y(pro)m(v)m(es)32 b(the)g(\014rst)e(assertion.)44 b(T)-8 b(o)32 b(pro)m(v)m(e)g(\()p Fp(ii)p Fq(\),)h(let)e Fp(g)s Fq(\()p Fp(\013)p Fq(\))e(=)e(0.)44 b(With)31 b(reference)h(to)g(\(2\))g(and)f(\(11\),)j(w)m(e)-24 3469 y(\014nd)148 3673 y Fp(\033)200 3687 y Fk(a)237 3668 y Ff(n)279 3687 y Fk(;)p Fl(0)338 3673 y Fq(\()p Fp(f)c(.)21 b(g)s Fq(\)\()p Fp(x)p Fq(\))27 b(=)840 3587 y Fj(Y)956 3604 y Fi(0)887 3777 y Fk(\013)1010 3599 y Fj(\000)1051 3673 y Fp(x)20 b Fn(\000)g Fp(a)1262 3635 y Fi(\000)p Fk(n)1364 3673 y Fp(f)10 b Fq(\()p Fp(\013)p Fq(\))1547 3599 y Fj(\001)1605 3673 y Fq(;)264 b(\()p Fp(\033)1981 3688 y Fk(a;b)2073 3673 y Fq(\()p Fp(g)s Fq(\)\)\()p Fp(x)p Fq(\))27 b(=)2469 3587 y Fj(Y)2585 3604 y Fi(0)2516 3777 y Fk(\013)2639 3599 y Fj(\000)2681 3673 y Fp(x)20 b Fn(\000)g Fp(a)2892 3635 y Fi(\000)p Fl(1)2986 3673 y Fq(\()p Fp(\013)h Fn(\000)f Fp(b)p Fq(\))3265 3599 y Fj(\001)3322 3673 y Fp(:)-24 3949 y Fq(from)30 b(whic)m(h)f(w)m(e)i(obtain)158 4154 y(\()p Fp(\033)245 4169 y Fk(a;b)337 4154 y Fq(\()p Fp(f)10 b Fq(\))20 b Fp(.)g(\033)599 4169 y Fk(a;b)691 4154 y Fq(\()p Fp(g)s Fq(\)\))d(\()p Fp(x)p Fq(\))41 b(=)1133 4067 y Fj(Y)1249 4084 y Fi(0)1180 4258 y Fk(\013)1303 4080 y Fj(\002)1340 4154 y Fp(x)21 b Fn(\000)e Fp(a)1551 4116 y Fi(\000)p Fk(n)1653 4154 y Fp(f)1723 4080 y Fj(\000)1765 4154 y Fp(aa)1861 4116 y Fi(\000)p Fl(1)1955 4154 y Fq(\()p Fp(\013)i Fn(\000)f Fp(b)p Fq(\))g(+)g Fp(b)2384 4080 y Fj(\001\003)2504 4154 y Fq(=)2615 4067 y Fj(Y)2732 4084 y Fi(0)2663 4258 y Fk(\013)2770 4154 y Fq(\()p Fp(x)h Fn(\000)e Fp(a)3016 4116 y Fi(\000)p Fk(n)3118 4154 y Fp(f)10 b Fq(\()p Fp(\013)p Fq(\)\))-24 4435 y(as)27 b(required.)38 b(P)m(art)27 b(\()p Fp(iii)p Fq(\))i(follo)m(ws)d(from)g (the)i(discriminan)m(t)c(form)m(ula)i(for)g(a)i(monic)e(p)s(olynomial)e (\([13)r(],)-24 4548 y(Chapter)30 b(1\))1236 4674 y(\001\()p Fp(f)10 b Fq(\))25 b(=)1649 4588 y Fj(Y)1558 4784 y Fl(1)p Fi(\024)p Fk(i)g Fq(0)g Fo(and)i(the)e(de)-5 b(gr)g(e)g(e)42 b(of)f Fp(g)j Fo(is)c(divisible)g(by)203 2208 y(char)p Fq(\()p Fp(K)7 b Fq(\))p Fo(.)24 2396 y Fq(\()p Fp(ii)p Fq(\))47 b Fo(If)37 b Fp(g)h Fq(=)33 b Fp(g)534 2363 y Fi(\003)531 2418 y Fk(a)612 2396 y Fo(for)38 b Fp(a)33 b Fn(6)p Fq(=)h(0)k Fo(is)f(a)h(r)-5 b(o)g(ot)39 b(of)f Fp(g)s Fo(,)h(the)e(de)-5 b(gr)g(e)g(e)39 b(of)e Fp(g)k Fo(is)d(divisible)f(by)g(the)h(multiplic)-5 b(ative)203 2509 y(or)g(der)34 b(of)f Fp(a)g Fo(in)f Fp(K)7 b Fo(.)-24 2821 y(Pr)-5 b(o)g(of.)84 b Fq(\()p Fp(i)p Fq(\))35 b(W)-8 b(e)35 b(\014rst)f(sho)m(w)g(that)g(if)f Fp(\013)p Fq(,)j Fp(\013)23 b Fq(+)f Fp(b)34 b Fq(and)g Fp(\014)39 b Fq(are)34 b(ro)s(ots)g(of)g Fp(g)s Fq(,)i(so)e(is)f Fp(\014)28 b Fq(+)23 b Fp(b)p Fq(.)51 b(Let)35 b Fp(\033)i Fq(b)s(e)c(an)-24 2934 y(elemen)m(t)38 b(of)f(the)h(Galois)f(group)f(of)i Fp(g)s Fq(,)i(sending)c Fp(\013)h Fq(to)i Fp(\014)5 b Fq(.)61 b(Then)37 b Fp(\014)30 b Fq(+)24 b Fp(b)37 b Fq(=)g Fp(\033)s Fq(\()p Fp(\013)p Fq(\))26 b(+)f Fp(b)37 b Fq(=)f Fp(\033)s Fq(\()p Fp(\013)26 b Fq(+)f Fp(b)p Fq(\),)-24 3047 y(sho)m(wing)k(that)h Fp(\014)24 b Fq(+)18 b Fp(b)30 b Fq(is)e(conjugate)j(to)g Fp(\013)19 b Fq(+)f Fp(b)p Fq(.)41 b(No)m(w,)30 b(let)g Fp(H)36 b Fq(b)s(e)29 b(the)h(collection)f(of)h(elemen)m(ts)g Fp(b)g Fq(of)f Fp(K)-24 3160 y Fq(for)h(whic)m(h)g Fp(\013)21 b Fq(+)f Fp(b)30 b Fq(is)g(a)h(ro)s(ot)g(of)g Fp(g)s Fq(.)42 b(Then)30 b Fp(H)j Fn(6)p Fq(=)25 b Fn(f)p Fq(0)p Fn(g)p Fq(,)33 b(b)m(y)d(h)m(yp)s(othesis,)g(and)g(it)g(is)g(an)h(additiv)m(e)e (group,)-24 3273 y(as)36 b(seen)g(from)g(rep)s(eated)g(applications)e (of)i(the)h(ab)s(o)m(v)m(e)g(argumen)m(t.)58 b(It)36 b(follo)m(ws)f(that)i(c)m(har\()p Fp(K)7 b Fq(\))35 b Fp(>)g Fq(0)-24 3386 y(and)27 b(that)i(the)f(subgroup)e Fn(h)15 b Fp(b)p Fn(i)28 b Fq(of)g Fp(H)35 b Fq(has)28 b(order)f Fp(p)e Fq(:=)g(c)m(har)q(\()p Fp(K)7 b Fq(\),)29 b(and)e(so)h(the)g(order)f(of)h Fp(H)35 b Fq(is)27 b(divisible)-24 3499 y(b)m(y)h Fp(p)p Fq(.)40 b(If)28 b Fp(\014)34 b Fq(is)27 b(another)i(ro)s(ot)g(of)f Fp(g)33 b Fq(not)28 b(of)h(the)g(form)f Fp(\013)17 b Fq(+)f Fp(b;)30 b(b)c Fn(2)f Fp(K)7 b Fq(,)28 b(then)g(the)h(corresp)s(onding)e(group)-24 3611 y(is)33 b(still)f Fp(H)7 b Fq(,)36 b(again)e(from)g(the)h(ab)s(o)m (v)m(e)g(argumen)m(t.)53 b(Rep)s(eating)34 b(this)g(pro)s(cedure)f(un)m (til)f(all)i(ro)s(ots)g(of)h Fp(g)-24 3724 y Fq(are)c(accoun)m(ted)g (for,)g(yields)d(the)j(result.)117 3837 y(\()p Fp(ii)p Fq(\))h(This)c(pro)s(of)i(is)f(the)i(m)m(ultiplicativ)m(e)d(v)m(ersion) i(of)g(that)h(giv)m(en)g(ab)s(o)m(v)m(e.)p 3446 3768 V 3446 3834 4 67 v 3516 3834 V 3446 3837 74 4 v -24 4180 a Fh(3.1)112 b(Self-image)37 b(of)h(blo)s(c)m(ks)-24 4352 y Fq(W)-8 b(e)23 b(consider)e(the)h(self-image)g(of)g(a)g(blo)s(c) m(k)f(\002)h(\(cf.)h(Equation)e(\(1\)\),)k(whic)m(h)c(consists)g(of)h (all)f(p)s(olynomials)-24 4465 y(of)32 b(the)g(t)m(yp)s(e)h Fp(f)d(.)22 b(f)643 4427 y Fl(+)633 4494 y Fk(b)701 4465 y Fq(,)33 b(with)e Fp(f)37 b Fn(2)28 b Fq(\002)k(and)f Fp(b)h Fq(a)h(non-zero)f(elemen)m(t)h(of)f Fp(K)7 b Fq(.)46 b(F)-8 b(rom)32 b(Lemma)g(3.1)i(\()p Fp(iv)s Fq(\),)g(it)-24 4578 y(follo)m(ws)c(that)j Fp(f)d(.)21 b(f)670 4540 y Fl(+)660 4607 y Fk(b)760 4578 y Fq(do)s(es)31 b(not)h(dep)s(end)e(on)h (the)h(c)m(hoice)g(of)g Fp(f)41 b Fq(in)30 b(\002.)44 b(But)31 b(then,)h(if)e(w)m(e)i(let)g Fn(B)s Fq(\()p Fp(f)10 b Fq(\))31 b(b)s(e)-24 4691 y(the)f(set)h(of)g(blo)s(c)m(ks)f (of)g(an)g(extension,)h(the)f(mapping)729 4895 y Fn(F)794 4910 y Fk(b)854 4895 y Fq(:)25 b Fn(B)j(!)d(B)335 b Fq(\002)1574 4910 y Fk(f)1659 4895 y Fn(7!)40 b Fq(\002)1861 4925 y Fk(f)7 b(.f)1978 4895 y Fe(+)1971 4949 y Ff(b)2366 4895 y Fq(0)26 b Fn(6)p Fq(=)e Fp(b)i Fn(2)f Fp(K)598 b Fq(\(15\))1725 5225 y(9)p eop %%Page: 10 10 10 9 bop -24 573 a Fq(is)29 b(w)m(ell-de\014ned.)39 b(F)-8 b(rom)31 b(Lemma)g(3.1)g(\()p Fp(i)p Fq(\))g(and)f(\()p Fp(ii)p Fq(\))i(w)m(e)f(\014nd)d(the)j(comm)m(utation)g(relation)1187 777 y Fp(\033)1239 791 y Fk(a)1276 772 y Ff(n)1319 791 y Fk(;)p Fl(0)1378 777 y Fq(\()p Fn(F)1478 792 y Fk(b)1513 777 y Fq(\(\002\)\))26 b(=)f Fn(F)1876 792 y Fk(ab)1948 777 y Fq(\()p Fp(\033)2035 791 y Fk(a;)p Fl(0)2132 777 y Fq(\(\002\)\))-24 981 y(where)30 b Fp(n)g Fq(is)f(the)i(degree)g(of)f (the)h(blo)s(c)m(k.)117 1094 y(By)37 b(means)f(of)h Fn(F)725 1109 y Fk(b)796 1094 y Fq(w)m(e)g(can)g(construct)g(an)g(orien)m(ted)f (graph,)i(whose)f(v)m(ertices)g(are)g(blo)s(c)m(ks,)h(and)-24 1207 y(where)i(t)m(w)m(o)i(blo)s(c)m(ks)d(\002)i(and)e(\002)1085 1174 y Fi(0)1149 1207 y Fq(are)i(joined)e(b)m(y)h(an)h(arc)f(if)g(\002) 2190 1174 y Fi(0)2253 1207 y Fq(b)s(elongs)f(to)j(the)e(self-image)h (of)f(\002.)-24 1320 y(This)35 b(graph,)i(called)f(the)h Fo(blo)-5 b(ck)38 b(gr)-5 b(aph,)40 b Fq(describ)s(es)35 b(a)h(v)m(ery)h(sp)s(eci\014c)f(but)f(signi\014can)m(t)h(class)g(of)h (self-)-24 1433 y(in)m(teractions.)73 b(W)-8 b(e)43 b(pro)m(v)m(e)f(t)m (w)m(o)h(results:)61 b(the)42 b(\014rst)e(pro)m(vides)h(a)h(su\016cien) m(t)e(condition)g(for)h(degree)-24 1546 y(in)m(v)-5 b(ariance)34 b(of)h(the)g(self-image)f(of)h(a)g(blo)s(c)m(k;)i(the)e(second)g(sho)m (ws)g(that)g(in)m(tersection)f(of)h(self-images)-24 1659 y(lead)30 b(to)h(in)m(v)-5 b(arian)m(ts.)-24 1871 y Fm(Prop)s(osition) 36 b(3.3)46 b Fo(L)-5 b(et)31 b Fq(\002)g Fo(b)-5 b(e)30 b(a)i(blo)-5 b(ck)31 b(of)g(de)-5 b(gr)g(e)g(e)32 b Fp(n)p Fo(,)e(wher)-5 b(e)32 b Fp(n)e Fo(is)h(prime)h(and)f(char)p Fq(\()p Fp(K)7 b Fq(\))27 b Fn(6)p Fq(=)e Fp(n)p Fo(.)41 b(Then)-24 1984 y(the)33 b(self-image)f(of)h Fq(\002)f Fo(has)i(de)-5 b(gr)g(e)g(e)34 b Fp(n)p Fo(.)-24 2197 y(Pr)-5 b(o)g(of.)84 b Fq(Let)35 b(\002)30 b(=)h(\002)739 2212 y Fk(f)784 2197 y Fq(,)k(and)e(let)h Fp(\013)g Fq(b)s(e)f(a)i(ro)s (ot)f(of)g Fp(f)10 b Fq(.)50 b(If)33 b Fp(b)e Fn(6)p Fq(=)g(0,)k(the)f(p)s(olynomial)d Fp(f)h(.)23 b(f)3097 2159 y Fl(+)3087 2226 y Fk(b)3189 2197 y Fq(has)34 b(ro)s(ot)-24 2310 y Fp(f)10 b Fq(\()p Fp(\013)c Fn(\000)g Fp(b)p Fq(\))25 b(=)g Fn(\000)p Fp(n)15 b(b)g(\013)655 2277 y Fk(n)p Fi(\000)p Fl(1)798 2310 y Fq(+)6 b Fn(\001)15 b(\001)g(\001)p Fq(,)25 b(whic)m(h)d(is)g(a)h(p)s(olynomial)e(in)h Fp(\013)h Fq(of)g(degree)h Fp(n)6 b Fn(\000)g Fq(1,)24 b(since,)g(b)m(y)f (assumption,)-24 2423 y Fp(b)15 b(n)27 b Fn(6)p Fq(=)h(0.)46 b(Because)33 b Fp(n)f Fq(is)f(prime,)g(the)h(degree)h(of)f Fp(f)10 b Fq(\()p Fp(\013)21 b Fn(\000)g Fp(b)p Fq(\))32 b(is)f(either)h Fp(n)f Fq(or)h(1.)46 b(In)31 b(the)h(latter)h(case,)g (if)-24 2535 y Fp(f)10 b Fq(\()p Fp(\013)18 b Fn(\000)g Fp(b)p Fq(\))25 b(=)g Fp(d)h Fn(2)f Fp(K)7 b Fq(,)29 b(then)g(w)m(e)h(ha)m(v)m(e)g Fp(f)10 b Fq(\()p Fp(\013)18 b Fn(\000)g Fp(b)p Fq(\))g Fn(\000)g Fp(d)26 b Fq(=)f(0,)30 b(whic)m(h)e(is)g(imp)s(ossible)e(since)i Fp(\013)i Fq(is)e(algebraic) -24 2648 y(of)i(degree)h Fp(n)p Fq(.)p 3446 2579 74 4 v 3446 2645 4 67 v 3516 2645 V 3446 2648 74 4 v -24 2961 a Fm(Prop)s(osition)36 b(3.4)46 b Fo(If)f(a)h(blo)-5 b(ck)45 b Fq(\002)g Fo(interse)-5 b(cts)46 b(its)f(self-image,)j(then)d Fq(\002)g Fo(c)-5 b(ontains)47 b(a)e(p)-5 b(olynomial)-24 3073 y(which)33 b(is)g(a)g(right-left)g(invariant)h(of)f(two)g(p)-5 b(olynomials)36 b(in)c(the)h(same)h(blo)-5 b(ck.)-24 3286 y(Pr)g(o)g(of.)84 b Fq(By)34 b(assumption,)g(a)g(p)s(olynomial)d Fp(f)43 b Fq(exists)33 b(suc)m(h)h(that)g Fp(f)e(.)23 b(f)2458 3248 y Fl(+)2448 3315 y Fk(b)2547 3286 y Fq(=)30 b Fp(f)2703 3253 y Fl(+)2693 3308 y Fk(c)2795 3286 y Fq(for)k(some)g Fp(b;)15 b(c)32 b Fn(2)e Fp(K)7 b Fq(,)-24 3399 y(with)29 b Fp(b)c Fn(6)p Fq(=)g(0.)41 b(Then,)30 b(letting)g Fp(\022)d Fq(=)e Fp(f)1226 3366 y Fl(+)1216 3421 y Fk(c)1285 3399 y Fq(,)30 b(from)g(Lemma)g(3.1)15 b(\()p Fp(iv)s Fq(\))33 b(w)m(e)e(obtain)1197 3603 y Fp(\022)1240 3618 y Fi(\000)p Fk(b)1349 3603 y Fp(.)20 b(\022)28 b Fq(=)d Fp(\022)275 b(\022)22 b(.)f(\022)2073 3618 y Fk(b)2132 3603 y Fq(=)k Fp(\022)s(;)1059 b Fq(\(16\))-24 3807 y(that)31 b(is,)f Fp(\022)i Fq(is)d(the)i(desired)e(righ)m(t-left) h(in)m(v)-5 b(arian)m(t)29 b(of)i(the)g(w)m(edge)g(op)s(erator.)p 3446 3738 V 3446 3804 4 67 v 3516 3804 V 3446 3807 74 4 v 117 4020 a(If,)j(with)e(the)h(ab)s(o)m(v)m(e)i(notation,)f(one)g (also)f(has)g(that)h Fp(\022)2022 4035 y Fk(b)2078 4020 y Fp(.)23 b(\022)32 b Fq(=)e Fp(\022)s Fq(,)j(then)g(the)g(self-in)m (tersection)g(of)-24 4133 y(a)g(blo)s(c)m(k)f(yields)f(t)m(w)m(o)j (stable)e(2-sets)i(of)e(t)m(yp)s(e)h(I)s(I)s(I,)e(namely)h Fn(f)p Fp(\022)s(;)15 b(\022)2214 4148 y Fi(\006)p Fk(b)2303 4133 y Fn(g)p Fq(.)48 b(One)32 b(v)m(eri\014es)g(that)h(this)e(is)h (the)-24 4246 y(case)g(precisely)d(when)h Fp(\022)s Fq(\()p Fp(\013)20 b Fq(+)g Fp(b)p Fq(\))31 b(and)g Fp(\022)s Fq(\()p Fp(\013)20 b Fn(\000)g Fp(b)p Fq(\))31 b(are)g(algebraic)g (conjugate,)h(whic)m(h)e(alw)m(a)m(ys)h(happ)s(ens)-24 4359 y(for)f(quadratic)g(p)s(olynomials)d(|)k(see)g(Section)f(4.1.)117 4472 y(An)f(instance)h(of)g(this)f(phenomenon)g(is)f(sho)m(wn)i(in)e (Figure)i(1.)41 b(The)29 b(quadratic)g(extension)h(graph)-24 4584 y(of)j Fm(F)148 4598 y Fl(3)221 4584 y Fq(consists)f(of)i(a)f (single)f(blo)s(c)m(k,)i(whic)m(h)e(in)m(tersects)h(\(indeed,)g(con)m (tains\))h(its)f(self-image,)h(due)e(to)-24 4697 y(Prop)s(osition)g (3.3,)37 b(giving)c(rise)g(to)i(the)g(righ)m(t-left)e(in)m(v)-5 b(arian)m(t)34 b Fp(g)s Fq(,)i(and)e(to)h(t)m(w)m(o)g(stable)f(2-sets)i (of)e(t)m(yp)s(e)-24 4810 y(I)s(I)s(I.)1702 5225 y(10)p eop %%Page: 11 11 11 10 bop -24 573 a Fr(4)135 b(Quadratic)45 b(self-in)l(teractions)-24 776 y Fq(Throughout)30 b(this)g(section,)i Fp(f)41 b Fq(denotes)32 b(a)f(quadratic)g(monic)g(irreducible)d(p)s(olynomial)g (o)m(v)m(er)33 b(a)f(\014eld)-24 889 y Fp(K)7 b Fq(,)39 b(with)e(discriminan)m(t)e(\001\()p Fp(f)10 b Fq(\).)63 b(W)-8 b(e)39 b(\014rst)e(c)m(haracterize)i(extension)f(and)f (transition)f(graphs,)k(and)-24 1002 y(construct)31 b(the)h(stable)f (2-sets.)44 b(Then)30 b(w)m(e)i(deriv)m(e)f(a)h(three-dimensional)c(sk) m(ew-map)k(describing)d(the)-24 1115 y(dynamics)g(of)h(2-sets,)i(with)d (whic)m(h)g(w)m(e)i(compute)g(p)s(erio)s(dic)d(p)s(oin)m(ts.)-24 1355 y Fh(4.1)112 b(Graphs)-24 1527 y Fq(The)30 b(quadratic)g (extension)g(graphs)f(ha)m(v)m(e)j(a)f(particularly)d(simple)g(form.) -24 1721 y Fm(Theorem)34 b(4.1)46 b Fo(A)d(quadr)-5 b(atic)46 b(extension)f(gr)-5 b(aph)45 b(c)-5 b(onsists)46 b(of)e(a)g(single)g (cluster,)j(whose)e(distinct)-24 1834 y(blo)-5 b(cks)42 b(have)g(distinct)g(discriminants.)70 b(If)41 b(char)p Fq(\()p Fp(K)7 b Fq(\))43 b Fn(6)p Fq(=)e(2)p Fo(,)j(then)e(the)g(blo) -5 b(ck-gr)g(aph)43 b(is)f(a)g(c)-5 b(omplete)-24 1946 y(gr)g(aph,)30 b(and)e(the)g(set)f(of)g(mappings)i Fn(F)1265 1961 y Fk(b)1300 1946 y Fo(,)f(de\014ne)-5 b(d)28 b(in)f(\(15\),)j (form)e(a)g(gr)-5 b(oup)28 b(of)f(p)-5 b(ermutations)30 b(of)e(blo)-5 b(cks,)-24 2059 y(isomorphic)37 b(to)e(the)g(multiplic)-5 b(ative)36 b(gr)-5 b(oup)36 b(of)f Fp(K)1715 2026 y Fl(2)1754 2059 y Fo(.)48 b(The)35 b(isomorphism)i(asso)-5 b(ciates)37 b Fp(b)3017 2026 y Fl(2)3085 2059 y Fn(2)29 b Fq(\()p Fp(K)3294 2026 y Fi(\003)3333 2059 y Fq(\))3368 2026 y Fl(2)3443 2059 y Fo(to)-24 2172 y(the)k(p)-5 b(ermutation)35 b(sending)e(the)g(blo)-5 b(ck)33 b Fq(\002)f Fo(to)h Fn(F)1618 2191 y Fk(b=)p Fl(2)1723 2172 y Fq(\(\002\))p Fo(.)117 2366 y Fq(The)e(completeness)g(of)g(the)h(blo)s(c)m(k)e(graph) h(means)g(that)h(the)f(diameter)g(of)g(the)h(extension)f(graph)-24 2479 y(is)j(at)j(most)f(t)m(w)m(o,)i(com)m(bining)c(p)s(olynomials)f (of)j(degree)g(1)g(and)f(2.)56 b(This)34 b(holds)g(in)g(a)i(strong)g (sense,)-24 2592 y(namely)e(for)g(ev)m(ery)i(quadratic)e(p)s (olynomials)d Fp(f)44 b Fq(and)34 b Fp(g)k Fq(there)d(exist)g(elemen)m (ts)f Fp(b)h Fq(and)f Fp(c)h Fq(in)e Fp(K)41 b Fq(suc)m(h)-24 2705 y(that)31 b(\()p Fp(f)263 2667 y Fl(+)253 2734 y Fk(b)342 2705 y Fp(.)20 b(f)10 b Fq(\))497 2672 y Fl(+)497 2727 y Fk(c)581 2705 y Fq(=)25 b Fp(g)s Fq(.)117 2818 y(W)-8 b(e)32 b(need)e(a)g(lemma:)-24 3012 y Fm(Lemma)j(4.2)46 b Fo(F)-7 b(or)33 b(every)g(quadr)-5 b(atic)34 b(p)-5 b(olynomial)36 b Fp(f)41 b Fo(and)34 b(every)e Fp(b)26 b Fn(2)f Fp(K)7 b Fo(,)32 b(the)h(fol)5 b(lowing)33 b(holds)56 3184 y Fq(\()p Fp(i)p Fq(\))46 b(\()p Fp(f)30 b(.)21 b(f)434 3146 y Fl(+)424 3213 y Fk(b)492 3184 y Fq(\)\()p Fp(x)p Fq(\))41 b(=)g(\()p Fp(x)20 b Fn(\000)g Fp(b)1039 3151 y Fl(2)1078 3184 y Fq(\))1113 3151 y Fl(2)1173 3184 y Fn(\000)g Fp(b)1303 3151 y Fl(2)1358 3184 y Fq(\001\()p Fp(f)10 b Fq(\))p Fo(.)24 3366 y Fq(\()p Fp(ii)p Fq(\))47 b(\001\()p Fp(f)30 b(.)20 b(f)509 3327 y Fl(+)499 3395 y Fk(b)568 3366 y Fq(\))41 b(=)f(\(2)15 b Fp(b)p Fq(\))924 3333 y Fl(2)964 3366 y Fq(\001\()p Fp(f)10 b Fq(\))p Fo(.)-7 3547 y Fq(\()p Fp(iii)p Fq(\))47 b Fo(If)33 b(char)p Fq(\()p Fp(K)7 b Fq(\))27 b Fn(6)p Fq(=)e(2)p Fo(,)33 b(then)g(the)g(p)-5 b(olynomials)36 b Fp(f)30 b(.)21 b(f)1898 3514 y Fl(+)1888 3569 y Fk(a)1989 3547 y Fo(and)33 b Fp(f)2220 3509 y Fl(+)2210 3576 y Fk(b)2299 3547 y Fp(.)21 b(f)2420 3509 y Fl(+)2410 3576 y Fk(b)p Fl(+)p Fk(c)2562 3547 y Fo(b)-5 b(elong)33 b(to)g(the)g(same)h(blo)-5 b(ck)203 3660 y(if)32 b(and)i(only)f(if)g Fp(a)25 b Fq(=)g Fn(\006)p Fp(c)p Fo(,)32 b(in)h(which)h(c)-5 b(ase)33 b(they)g(c)-5 b(oincide.)-24 3854 y(Pr)g(o)g(of.)84 b Fq(The)33 b(\014rst)f(t)m(w)m(o)j(iden)m(tities)d(are)h(v)m(eri\014ed)g (b)m(y)g(direct)f(calculation.)49 b(T)-8 b(o)33 b(pro)m(v)m(e)h(\()p Fp(iii)p Fq(\),)i(w)m(e)e(\014rst)-24 3967 y(reduce)h(it)g(to)h(the)g (case)h Fp(b)d Fq(=)f(0,)38 b(from)d(Lemma)g(3.1)15 b(\()p Fp(ii)p Fq(\).)59 b(Then)35 b(from)g(\()p Fp(i)p Fq(\))h(ab)s(o)m(v)m (e,)i(w)m(e)e(ha)m(v)m(e)h(that)f(if)-24 4080 y Fp(f)29 b(.)21 b(f)171 4047 y Fl(+)161 4102 y Fk(a)260 4080 y Fq(and)29 b Fp(f)491 4041 y Fl(+)481 4109 y Fk(b)570 4080 y Fp(.)21 b(f)691 4041 y Fl(+)681 4109 y Fk(b)p Fl(+)p Fk(c)830 4080 y Fq(b)s(elong)30 b(to)h(the)f(same)h(blo)s(c)m (k,)f(then)g(for)g(some)h Fp(d)26 b Fn(2)f Fp(K)826 4277 y Fq(\()p Fp(x)c Fq(+)e Fp(d)i Fn(\000)f Fp(a)1231 4240 y Fl(2)1270 4277 y Fq(\))1305 4240 y Fl(2)1365 4277 y Fn(\000)g Fp(a)1504 4240 y Fl(2)1559 4277 y Fq(\001\()p Fp(f)10 b Fq(\))40 b(=)g(\()p Fp(x)21 b Fn(\000)f Fp(c)2149 4240 y Fl(2)2188 4277 y Fq(\))2223 4240 y Fl(2)2283 4277 y Fn(\000)g Fp(c)2413 4240 y Fl(2)2468 4277 y Fq(\001\()p Fp(f)10 b Fq(\))-24 4463 y(whic)m(h)35 b(implies)f(that)j Fp(d)f Fq(=)f(0)i(and)f Fp(a)1262 4430 y Fl(2)1338 4463 y Fq(=)f Fp(c)1483 4430 y Fl(2)1523 4463 y Fq(.)59 b(Con)m(v)m(ersely) -8 b(,)39 b(if)c Fp(a)h Fq(=)f Fn(\006)p Fp(c)p Fq(,)k(from)d(part)g (\()p Fp(i)p Fq(\))i(ab)s(o)m(v)m(e)g(and)-24 4576 y(Lemma)30 b(3.1)15 b(\()p Fp(ii)p Fq(\),)34 b(w)m(e)c(ha)m(v)m(e)i(that)f Fp(f)e(.)21 b(f)1356 4543 y Fl(+)1346 4598 y Fk(a)1440 4576 y Fq(=)k Fp(f)1591 4538 y Fl(+)1581 4605 y Fk(b)1669 4576 y Fp(.)c(f)1790 4538 y Fl(+)1780 4605 y Fk(b)p Fi(\006)p Fk(a)1931 4576 y Fq(=)k Fp(f)2082 4538 y Fl(+)2072 4605 y Fk(b)2161 4576 y Fp(.)20 b(f)2281 4538 y Fl(+)2271 4605 y Fk(b)p Fl(+)p Fk(c)2391 4576 y Fq(.)p 3446 4506 74 4 v 3446 4573 4 67 v 3516 4573 V 3446 4576 74 4 v -24 4863 a Fo(Pr)-5 b(o)g(of)34 b(of)e(The)-5 b(or)g(em)34 b(4.1)122 b Fq(If)29 b Fp(\013)i Fq(is)e(a)h(ro)s(ot)g(of)g Fp(f)39 b Fq(and)30 b Fp(g)j Fq(is)c(an)h(irreducible)d(p)s(olynomial)g (in)i(the)h(same)-24 4976 y(extension,)f(then)g(the)g(ro)s(ots)h(of)f Fp(g)j Fq(are)e(linear)d(expressions)h(in)g Fp(\013)h Fq(with)f(co)s(e\016cien)m(ts)i(in)e Fp(K)7 b Fq(.)40 b(Th)m(us,)28 b(for)1702 5225 y(11)p eop %%Page: 12 12 12 11 bop -24 573 a Fq(some)31 b Fp(a;)15 b(b)27 b Fn(2)e Fp(K)7 b Fq(,)31 b(w)m(e)g(ha)m(v)m(e)h(that)g Fp(\033)1178 588 y Fk(a;b)1269 573 y Fq(\()p Fp(f)10 b Fq(\))26 b(=)g Fp(g)s Fq(,)32 b(i.e.,)f(there)g(is)f(a)h(single)e(cluster.)42 b(No)m(w,)32 b(from)e(Lemma)-24 686 y(3.1)15 b(\()p Fp(iii)p Fq(\))41 b(w)m(e)d(ha)m(v)m(e)i(\001\()p Fp(\033)832 701 y Fk(a;b)923 686 y Fq(\()p Fp(f)10 b Fq(\)\))54 b(=)f(\001\()p Fp(f)10 b Fq(\))p Fp(=a)1555 653 y Fl(2)1595 686 y Fq(,)40 b(so)e(distinct)f(v)-5 b(alues)37 b(of)h Fp(a)2553 653 y Fl(2)2631 686 y Fq(corresp)s(ond)f(to)i(distinct)-24 799 y(discriminan)m(ts.)f(No)m(w)31 b(let)f Fp(S)35 b Fq(b)s(e)30 b(the)h(sum)e(of)i(the)f(ro)s(ots)h(of)f Fp(f)10 b Fq(\()p Fp(x)p Fq(\).)41 b(One)30 b(v)m(eri\014es)f(that)942 1008 y Fp(\033)994 1023 y Fk(a;b)1085 1008 y Fq(\()p Fp(f)10 b Fq(\))26 b(=)f Fp(\033)1384 1022 y Fl(1)p Fk(;c)1473 1008 y Fq(\()p Fp(\033)1560 1023 y Fi(\000)p Fk(a;b)1707 1008 y Fq(\()p Fp(f)10 b Fq(\)\))249 b Fp(c)26 b Fq(=)2287 947 y(2)p Fp(b)20 b Fn(\000)g Fp(S)p 2287 988 257 4 v 2391 1071 a(a)-24 1201 y Fq(whic)m(h)47 b(sho)m(ws)i(that)g Fp(\033)801 1216 y Fk(a;b)892 1201 y Fq(\()p Fp(f)10 b Fq(\))49 b(and)f Fp(\033)1313 1216 y Fi(\000)p Fk(a;b)1459 1201 y Fq(\()p Fp(f)10 b Fq(\))48 b(b)s(elong)g(to)h(the)g(same)g(blo)s (c)m(k,)k(and)48 b(so)h(blo)s(c)m(ks)f(are)-24 1314 y(parametrized)30 b(b)m(y)g(discriminan)m(ts.)117 1427 y(Let)g(c)m(har\()p Fp(K)7 b Fq(\))26 b Fn(6)p Fq(=)f(2.)41 b(T)-8 b(o)29 b(sho)m(w)h(that)f(the)h(blo)s(c)m(k)f(graph)f(is)h(complete,)h(w)m(e)g (consider)e(the)h(equation)1306 1594 y Fn(F)1371 1609 y Fk(b)1406 1594 y Fq(\(\002)1512 1609 y Fk(f)1557 1594 y Fq(\))d(=)f(\002)1785 1624 y Fk(f)7 b(.f)1902 1595 y Fe(+)1895 1648 y Ff(b)1982 1594 y Fq(=)25 b(\002)2149 1608 y Fk(g)-24 1779 y Fq(where)33 b Fp(f)43 b Fq(and)34 b Fp(g)j Fq(are)d(giv)m(en)g(quadratic)f(irreducible)e(p)s(olynomials,) h(with)h Fp(E)5 b Fq(\()p Fp(f)10 b Fq(\))31 b(=)g Fp(E)5 b Fq(\()p Fp(g)s Fq(\).)52 b(W)-8 b(e)36 b(lo)s(ok)-24 1892 y(at)42 b(discriminan)m(ts.)69 b(Because)42 b(our)e(extension)h (is)f(separable,)j(and)d(all)g(discriminan)m(ts)e(are)j(square)-24 2005 y(m)m(ultiples)32 b(of)i(the)h(\014eld)e(discriminan)m(t,)g(w)m(e) i(ha)m(v)m(e)g(\001\()p Fp(g)s Fq(\))e(=)f Fp(k)2124 1972 y Fl(2)2164 2005 y Fq(\001\()p Fp(f)10 b Fq(\),)35 b(for)f(some)h(0)e Fn(6)p Fq(=)e Fp(k)36 b Fn(2)31 b Fp(K)7 b Fq(.)53 b(This,)-24 2117 y(together)38 b(with)e(Lemma)h(4.2\() p Fp(ii)p Fq(\),)k(giv)m(es)c(the)g(equation)g Fp(k)2019 2084 y Fl(2)2095 2117 y Fq(=)e(\(2)p Fp(b)p Fq(\))2355 2084 y Fl(2)2396 2117 y Fq(,)k(whic)m(h)c(can)i(b)s(e)g(solv)m(ed)f (for)h Fp(b)p Fq(,)-24 2230 y(since)27 b(c)m(har\()p Fp(K)7 b Fq(\))26 b Fn(6)p Fq(=)f(2.)40 b(Matc)m(hing)29 b(discriminan)m(ts)c(su\016ces,)j(since)f(w)m(e)i(ha)m(v)m(e)g(seen)f (that)g(discriminan)m(ts)-24 2343 y(iden)m(tify)h(blo)s(c)m(ks.)117 2456 y(W)-8 b(e)44 b(sho)m(w)f(that)h Fn(F)799 2471 y Fk(b)876 2456 y Fq(is)e(injectiv)m(e.)79 b(If)42 b(\002)h(and)g(\002) 1893 2423 y Fi(0)1959 2456 y Fq(are)g(distinct)e(blo)s(c)m(ks,)46 b(they)d(ha)m(v)m(e)i(distinct)-24 2569 y(discriminan)m(ts,)27 b(\001)k(and)e(\001)919 2536 y Fi(0)942 2569 y Fq(,)i(sa)m(y)-8 b(.)42 b(But)30 b(then)924 2736 y(\001\()p Fn(F)1100 2751 y Fk(b)1135 2736 y Fq(\(\002)1241 2751 y Fk(f)1286 2736 y Fq(\)\))c(=)f(2)p Fp(b)1562 2699 y Fl(2)1602 2736 y Fq(\001)g Fn(6)p Fq(=)g(2)p Fp(b)1883 2699 y Fl(2)1923 2736 y Fq(\001)1999 2699 y Fi(0)2047 2736 y Fq(=)g(\001\()p Fn(F)2319 2751 y Fk(b)2354 2736 y Fq(\(\002)2460 2750 y Fk(g)2500 2736 y Fq(\)\))-24 2903 y(whic)m(h)31 b(sho)m(ws)h(that)h Fn(F)765 2918 y Fk(b)800 2903 y Fq(\(\002\))c Fn(6)p Fq(=)g Fn(F)1135 2918 y Fk(b)1169 2903 y Fq(\(\002)1275 2870 y Fi(0)1299 2903 y Fq(\),)k(as)g(desired.)45 b(T)-8 b(o)33 b(pro)m(v)m(e)h(surjectivit)m(y)-8 b(,)33 b(w)m(e)f(m)m(ust)h (solv)m(e)g(for)f Fp(f)-24 3016 y Fq(the)c(equation)h Fn(F)565 3031 y Fk(b)599 3016 y Fq(\(\002)705 3031 y Fk(f)751 3016 y Fq(\))c(=)g(\002)978 3030 y Fk(g)1018 3016 y Fq(,)k(for)f(giv)m(en)g Fp(b)h Fq(and)e Fp(g)s Fq(.)41 b(F)-8 b(rom)29 b(what)f(w)m(as)h(pro)m(v)m(ed)f(ab)s(o)m(v)m (e,)j(this)c(amoun)m(ts)-24 3129 y(to)k(\014nd)e Fp(f)39 b Fq(suc)m(h)30 b(that)h(\001\()p Fp(f)10 b Fq(\)\(2)p Fp(b)p Fq(\))1110 3096 y Fl(2)1176 3129 y Fq(=)25 b(\001\()p Fp(g)s Fq(\);)32 b(since)d(c)m(har\()p Fp(K)7 b Fq(\))26 b Fn(6)p Fq(=)f(2,)31 b(w)m(e)g(can)g(tak)m(e)h Fp(f)i Fq(=)25 b Fp(\033)3014 3144 y Fl(2)p Fk(b;)p Fl(0)3139 3129 y Fq(\()p Fp(g)s Fq(\).)117 3242 y(Th)m(us,)33 b(if)f(c)m(har\()p Fp(K)7 b Fq(\))30 b Fn(6)p Fq(=)g(2,)k(eac)m(h)g(non-zero)g(v)-5 b(alue)32 b(of)i Fp(b)f Fq(de\014nes)f(a)h(p)s(erm)m(utation)f(of)h (blo)s(c)m(ks.)48 b(No)m(w)-24 3355 y(consider)29 b(the)i(mapping)1113 3468 y Fp(\026)25 b Fq(:)41 b Fp(b)1298 3430 y Fl(2)1363 3468 y Fn(7!)25 b(F)1544 3486 y Fk(b=)p Fl(2)1981 3468 y Fq(0)h Fn(6)p Fq(=)f Fp(b)g Fn(2)g Fp(K)-24 3616 y Fq(sending)c(\()p Fp(K)416 3583 y Fi(\003)455 3616 y Fq(\))490 3583 y Fl(2)552 3616 y Fq(to)i(the)f(symmetric)g(group)g(on)g Fn(B)s Fq(.)37 b(The)22 b(c)m(hoice)h(of)f Fp(b)g Fq(among)h(the)g (square)f(ro)s(ots)g(of)g Fp(b)3396 3583 y Fl(2)3458 3616 y Fq(is)-24 3729 y(irrelev)-5 b(an)m(t,)23 b(due)e(to)h(Lemma)g (4.2\()p Fp(iii)p Fq(\).)41 b(F)-8 b(rom)22 b(the)g(same)g(lemma,)h (part)f(\()p Fp(ii)p Fq(\),)j(w)m(e)d(see)h(that)f Fp(\026)f Fq(asso)s(ciates)-24 3842 y(to)36 b Fp(b)131 3809 y Fl(2)206 3842 y Fq(the)g(p)s(erm)m(utation)f(sending)f(the)i(blo)s(c)m(k)f(of)h (discriminan)m(t)c(\001)k(to)g(that)g(of)g(discriminan)m(t)d Fp(b)3379 3809 y Fl(2)3418 3842 y Fq(\001.)-24 3955 y(Keeping)c(this)h (in)f(mind,)f(w)m(e)j(\014nd)e(that)891 4122 y Fp(\026)p Fq(\(\()p Fp(bc)p Fq(\))1129 4084 y Fl(2)1169 4122 y Fq(\))d(=)f Fn(F)1391 4140 y Fk(bc=)p Fl(2)1552 4122 y Fq(=)g Fn(F)1713 4140 y Fk(b=)p Fl(2)1839 4122 y Fn(\016)20 b(F)1969 4140 y Fk(c=)p Fl(2)2100 4122 y Fq(=)25 b Fp(\026)p Fq(\()p Fp(b)2325 4084 y Fl(2)2365 4122 y Fq(\))p Fp(\026)p Fq(\()p Fp(c)2529 4084 y Fl(2)2569 4122 y Fq(\))-24 4289 y(e.g.,)37 b Fp(\026)e Fq(is)f(a)h(group)f(homomorphism.)52 b(Its)35 b(k)m(ernel)g(is)f(trivial,)g Fp(\026)p Fq(\(1)2336 4256 y Fl(2)2376 4289 y Fq(\))f(=)f Fn(F)2612 4307 y Fl(1)p Fk(=)p Fl(2)2723 4289 y Fq(,)k(and)e(so)h Fp(\026)g Fq(de\014nes)f(a)-24 4402 y(faithful)28 b(action)j(of)f(\()p Fp(K)789 4369 y Fi(\003)829 4402 y Fq(\))864 4369 y Fl(2)934 4402 y Fq(on)g Fn(B)s Fq(.)p 3446 4332 74 4 v 3446 4398 4 67 v 3516 4398 V 3446 4401 74 4 v 117 4589 a(Thanks)i(to)i(Prop)s (osition)d(3.4,)k(the)f(iden)m(tit)m(y)e(p)s(erm)m(utation)g Fn(F)2313 4608 y Fl(1)p Fk(=)p Fl(2)2457 4589 y Fq(maps)h(the)g(whole)f (blo)s(c)m(k)h(to)h(a)-24 4702 y(righ)m(t-left)c(in)m(v)-5 b(arian)m(t)29 b(of)i(the)g(w)m(edge)g(op)s(erator,)g(whic)m(h,)e(from) h(Lemma)g(4.2\()p Fp(ii)p Fq(\),)k(is)29 b(giv)m(en)h(b)m(y)1299 4936 y Fp(\022)s Fq(\()p Fp(x)p Fq(\))25 b(=)1588 4807 y Fj(\022)1655 4936 y Fp(x)20 b Fn(\000)1828 4874 y Fq(1)p 1828 4915 46 4 v 1828 4998 a(4)1883 4807 y Fj(\023)1950 4830 y Fl(2)2010 4936 y Fn(\000)2111 4874 y Fq(\001)p 2111 4915 76 4 v 2126 4998 a(4)3358 4936 y(\(17\))1702 5225 y(12)p eop %%Page: 13 13 13 12 bop -24 573 a Fq(where)33 b(\001)h(is)e(the)i(blo)s(c)m(k)g (discriminan)m(t.)48 b(F)-8 b(rom)34 b(Lemma)g(4.2\()p Fp(iii)p Fq(\),)j(this)c(p)s(olynomial)e(is)i(unique:)46 b(w)m(e)-24 686 y(call)31 b(it)f(the)i Fo(c)-5 b(entr)g(e)39 b Fq(of)32 b(the)f(blo)s(c)m(k.)43 b(Because)33 b Fp(\022)c Fq(=)e Fp(\022)c(.)e(\022)1904 648 y Fl(+)1901 719 y(1)p Fk(=)p Fl(2)2042 686 y Fq(=)27 b Fp(\022)c(.)f(\022)2319 648 y Fl(+)2316 719 y Fi(\000)p Fl(1)p Fk(=)p Fl(2)2507 686 y Fq(=)27 b Fp(\022)2648 704 y Fl(1)p Fk(=)p Fl(2)2778 686 y Fp(.)22 b(\022)33 b Fq(=)27 b Fp(\022)3062 704 y Fi(\000)p Fl(1)p Fk(=)p Fl(2)3247 686 y Fp(.)22 b(\022)s Fq(,)31 b(w)m(e)-24 799 y(obtain)f(the)g(follo)m(wing)f(t)m(yp)s(e)i(I) s(I)s(I)d(stable)j(sets)897 1003 y Fn(f)p Fp(\022)s(;)15 b(\022)1074 965 y Fl(+)1071 1036 y(1)p Fk(=)p Fl(2)1180 1003 y Fn(g)250 b(f)p Fp(\022)s(;)15 b(\022)1652 965 y Fl(+)1649 1036 y Fi(\000)p Fl(1)p Fk(=)p Fl(2)1813 1003 y Fn(g)250 b Fq(c)m(har\()p Fp(K)7 b Fq(\))26 b Fn(6)p Fq(=)f(2)760 b(\(18\))-24 1207 y(whic)m(h)30 b(are)i(in)e (bi-unique)e(corresp)s(ondence)j(with)f(the)h(blo)s(c)m(k)g (discriminan)m(ts)d(\001)f Fn(2)f Fp(K)7 b Fq(.)44 b(An)30 b(example)-24 1320 y(is)35 b(giv)m(en)g(in)g(Figure)g(1,)i(with)e Fp(g)j Fq(=)33 b Fp(\022)s Fq(,)k Fp(h)d Fq(=)g Fp(\022)1566 1339 y Fl(1)p Fk(=)p Fl(2)1676 1320 y Fq(,)j(and)e Fp(f)43 b Fq(=)34 b Fp(\022)2156 1339 y Fi(\000)p Fl(1)p Fk(=)p Fl(2)2321 1320 y Fq(.)56 b(\(Note)38 b(ho)m(w)m(ev)m(er,)g(that)f(the)f (set)-24 1442 y Fn(f)p Fp(\022)s(;)15 b(\022)153 1404 y Fl(+)150 1475 y(1)p Fk(=)p Fl(2)260 1442 y Fp(;)g(\022)346 1404 y Fl(+)343 1475 y Fi(\000)p Fl(1)p Fk(=)p Fl(2)508 1442 y Fn(g)p Fq(,)37 b(whic)m(h)d(is)h(stable)g(in)f(Figure)h(1,)j(is) c(not)i(stable)f(in)g(general.\))56 b(If)35 b Fp(K)42 b Fq(is)35 b(an)g(in\014nite)-24 1555 y(\014eld,)i(then)f(the)h(n)m(um) m(b)s(er)e(of)i(blo)s(c)m(ks)f(of)h(an)m(y)g(non-trivial)d(extension)i (is)g(in\014nite,)g(and)g(so)h(w)m(e)g(ha)m(v)m(e)-24 1668 y(in\014nitely)27 b(man)m(y)k(quadratic)e(stable)i(2-sets)g(of)g (t)m(yp)s(e)f(I)s(I)s(I.)117 1781 y(In)h(the)i(follo)m(wing)e(table,)h (w)m(e)h(displa)m(y)e(all)g(parametrized)h(families)e(of)i(quadratic)g (stable)g(2-sets,)-24 1893 y(for)e(c)m(har\()p Fp(K)7 b Fq(\))26 b Fn(6)p Fq(=)f(2.)995 2106 y(T)p Fd(ABLE)30 b Fq(I:)h Fd(QUADRA)-6 b(TIC)23 b(ST)-6 b(ABLE)30 b Fq(2-)p 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y(Eac)m(h)j(blo)s(c)m(k)f(con)m(tains)h(at)g(most)f Fp(q)k Fq(p)s(olynomials,)31 b(but)i(it)g(ma)m(y)h(ev)m(en)g(consist)f(of)g(a) h(single)e(p)s(oly-)-24 1778 y(nomial,)52 b(namely)c(an)g(a\016ne)h Fp(q)s Fq(-p)s(olynomial)d(o)m(v)m(er)k Fm(F)1901 1792 y Fk(q)1988 1778 y Fq(\(see)f([13)r(],)k(Chapter)48 b(3.4\).)97 b(Ho)m(w)m(ev)m(er,)55 b(if)-24 1891 y(gcd\()p Fp(q)s(;)15 b(n)p Fq(\))48 b(=)f(1,)h(from)43 b(Lemma)h(3.2)15 b(\()p Fp(i)p Fq(\))46 b(w)m(e)e(see)g(that)h(no)e(irreducible)d(p)s (olynomial)h(of)j(degree)h Fp(n)-24 2004 y Fq(o)m(v)m(er)37 b Fm(F)242 2018 y Fk(q)315 2004 y Fq(can)f(ha)m(v)m(e)h(ro)s(ots)f (di\013ering)e(b)m(y)h(elemen)m(ts)i(of)e Fm(F)1983 2018 y Fk(q)2021 2004 y Fq(.)57 b(So)36 b(if)e Fp(f)44 b Fn(2)33 b Fp(E)5 b Fq(\()p Fp(q)2657 1971 y Fk(n)2705 2004 y Fq(\),)38 b(the)d Fp(q)k Fq(p)s(olynomials)-24 2117 y Fp(f)31 2084 y Fl(+)21 2140 y Fk(a)89 2117 y Fp(;)31 b(a)25 b Fn(2)g Fm(F)370 2131 y Fk(q)438 2117 y Fq(are)31 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Fq(The)34 b(case)i Fp(q)f Fq(=)c(3)k(is)f(trivial,)f(so)i(w)m(e)g(assume)f Fp(q)h(>)d Fq(3.)53 b(Consider)33 b(the)h(p)s(olynomial)e Fp(L)p Fq(\()p Fp(x)p Fq(\))g(=)-24 3645 y Fp(F)13 b Fq(\()p Fp(x)p Fq(\))i Fp(G)p Fq(\()p Fp(x)p Fq(\))g Fp(H)7 b Fq(\()p Fp(x)p Fq(\))32 b(where)-24 3849 y Fp(F)13 b Fq(\()p Fp(x)p Fq(\))26 b(=)f Fp(x)343 3811 y Fl(\()p Fk(q)r Fi(\000)p Fl(1\))p Fk(=)p Fl(2)616 3849 y Fn(\000)19 b Fq(\()p Fp(x)h Fn(\000)f Fp(a)p Fq(\))986 3811 y Fl(\()p Fk(q)r Fi(\000)p Fl(1\))p Fk(=)p Fl(2)1240 3849 y Fq(;)140 b Fp(G)p Fq(\()p Fp(x)p Fq(\))26 b(=)e Fp(x)1772 3811 y Fl(\()p Fk(q)r Fi(\000)p Fl(1\))p Fk(=)p Fl(2)2046 3849 y Fq(+)19 b(\()p Fp(x)h Fn(\000)f Fp(a)p Fq(\))2416 3811 y Fl(\()p Fk(q)r Fi(\000)p Fl(1\))p Fk(=)p Fl(2)2670 3849 y Fq(;)140 b Fp(H)7 b Fq(\()p Fp(x)p Fq(\))26 b(=)f Fp(x)p Fq(\()p Fp(x)19 b Fq(+)h Fp(a)p Fq(\))p Fp(:)-24 4053 y Fq(F)-8 b(rom)31 b(Euler)f(criterion,)g(for)h(an)m(y)g Fp(x)26 b Fn(2)g Fm(F)1393 4067 y Fk(q)1431 4053 y Fq(,)31 b(precisely)f(one)h(of)g Fp(F)13 b Fq(\()p Fp(x)p Fq(\),)32 b Fp(G)p Fq(\()p Fp(x)p Fq(\),)f Fp(H)7 b Fq(\()p Fp(x)p Fq(\))32 b(is)e(zero.)43 b(F)-8 b(urther-)-24 4166 y(more,)37 b(deg)p Fp(F)48 b Fq(=)33 b(\()p Fp(q)27 b Fn(\000)d Fq(3\))p Fp(=)p Fq(2,)38 b(deg)p Fp(G)c Fq(=)g(\()p Fp(q)27 b Fn(\000)c Fq(1\))p Fp(=)p Fq(2,)39 b(deg)p Fp(H)i Fq(=)34 b(2,)k(and)d(hence)g(deg)p Fp(L)g Fq(=)e Fp(q)s Fq(,)k(so)f(that)g Fp(L)-24 4279 y Fq(splits)22 b(completely)j(o)m(v)m(er)g Fm(F)914 4293 y Fk(q)977 4279 y Fq(and)f Fp(L)g Fq(is)g(a)g(constan)m (t)i(m)m(ultiple)c(of)j Fp(x)2250 4246 y Fk(q)2297 4279 y Fn(\000)8 b Fp(x)p Fq(.)39 b(So)24 b(the)h(\()p Fp(q)12 b Fn(\000)c Fq(3\))p Fp(=)p Fq(2)26 b(ro)s(ots)e(of)h Fp(F)-24 4392 y Fq(are)k(the)g(v)-5 b(alues)29 b(of)g Fp(x)g Fq(for)f(whic)m(h)g Fp(x)h Fq(and)f Fp(x)18 b Fq(+)f Fp(a)29 b Fq(are)g(b)s(oth)f(non-zero,)j(and)d(ha)m(v)m(e)i(the) f(same)h(quadratic)-24 4505 y(c)m(haracter.)117 4618 y(Consider)36 b(the)i(in)m(v)m(olution)e Fp(x)h 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Fq(,)35 b Fp(x)-24 686 y Fq(and)30 b Fp(x)20 b Fq(+)g Fp(a)30 b Fq(are)h(sim)m(ultaneously)d(non-squares.) p 3446 616 74 4 v 3446 682 4 67 v 3516 682 V 3446 685 74 4 v 117 898 a(Applying)35 b(the)j(lemma)f(with)f Fp(a)h Fq(=)g Fp(b)1428 865 y Fl(2)1492 898 y Fn(\000)25 b Fp(c)1627 865 y Fl(2)1667 898 y Fq(,)39 b Fp(x)e Fq(=)g Fp(c)1967 865 y Fl(2)2031 898 y Fn(\000)25 b Fq(4)p Fp(r)s Fq(,)39 b(and)e Fp(b)2503 865 y Fl(2)2580 898 y Fn(6)p Fq(=)g Fp(c)2727 865 y Fl(2)2766 898 y Fq(,)j(w)m(e)e(conclude)f(that)-24 1011 y(there)29 b(are)g(\()p Fp(q)20 b Fn(\000)c Fq(3\))p Fp(=)p Fq(4)31 b(v)-5 b(alues)28 b(of)g Fp(r)k Fq(for)c(whic)m(h)f(b)s (oth)h(p)s(olynomials)e Fp(x)2348 978 y Fl(2)2404 1011 y Fq(+)17 b Fp(bx)g Fq(+)f Fp(r)31 b Fq(and)d Fp(x)2986 978 y Fl(2)3043 1011 y Fq(+)16 b Fp(cx)h Fq(+)g Fp(r)31 b Fq(are)-24 1124 y(irreducible.)36 b(\(Plainly)-8 b(,)25 b(if)g Fp(b)939 1091 y Fl(2)1004 1124 y Fq(=)g Fp(c)1139 1091 y Fl(2)1179 1124 y Fq(,)i(there)f(are)g(\()p Fp(q)15 b Fn(\000)c 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(en)m(tually)h(p)s(erio)s(dic,)f(and)h(a)g(natural)f(problem)g(is)g(to) h(determine)f(the)i(structure)-24 3009 y(of)37 b(p)s(erio)s(dic)d(sets) j(o)m(v)m(er)h(a)f(giv)m(en)f(\014nite)g(\014eld.)58 b(This)35 b(problem)g(ma)m(y)i(w)m(ell)f(b)s(e)g(in)m(tractable,)j (that)e(is,)-24 3122 y(not)28 b(solv)-5 b(able)27 b(in)f(p)s(olynomial) f(time.)40 b(Sp)s(eci\014cally)-8 b(,)26 b(ev)m(en)i(in)f(its)g (simplest)f(form)h(|)g(determining)f(the)-24 3235 y(p)s(erio)s(ds)e(of) k(quadratic)e(2-sets)i(|)f(this)f(problem)f(seems)j(at)f(least)h(as)f (di\016cult)e(as)i(that)h(of)f(computing)-24 3348 y(the)i(p)s(erio)s(d) e(of)i(the)h(doubling)c(map)j(o)m(v)m(er)i(a)e(\014nite)f(\014eld)g (\(cf.)i(commen)m(t)g(at)g(the)g(end)e(of)h(Section)g(4.2\),)-24 3461 y(whic)m(h)g(requires)g(kno)m(wledge)h(of)h(the)f(discrete)g (logarithm)g([12)q(].)1702 5225 y(20)p eop %%Page: 21 21 21 20 bop -24 573 a Fr(References)21 776 y Fq([1])47 b(T.)34 b(M.)h(Ap)s(ostol,)g(In)m(tro)s(duction)e(to)i(analytic)f(n)m (um)m(b)s(er)f(theory)-8 b(,)36 b(Springer)c(V)-8 b(erlag,)36 b(New)f(Y)-8 b(ork)163 889 y(\(1976\).)21 1076 y([2])47 b(T.)30 b(Bousc)m(h,)i(Sur)e(quelques)f(probl)m(\022)-43 b(emes)31 b(de)g(la)f(dynamique)f(holomorphe,)h(Ph.D.)i(thesis,)e(Uni-) 163 1189 y(v)m(ersit)m(\023)-43 b(e)31 b(de)g(P)m(aris-Sud,)d(Cen)m (tre)j(d'Orsa)m(y)f(\(1992\).)21 1377 y([3])47 b(A.)37 b(Batra)h(and)f(P)-8 b(.)37 b(Morton,)j(Algebraic)c(dynamics)g(of)h(p)s (olynomial)e(maps)h(on)h(the)g(algebraic)163 1490 y(closure)30 b(of)g(a)h(\014nite)e(\014eld)g(I,)i Fb(Ro)s(c)m(ky)g(Moun)m(tain)f(J.) g(of)g(Math.)i Fm(24)f Fq(\(1994\))h(453{481.)21 1677 y([4])47 b(A.)37 b(Batra)h(and)f(P)-8 b(.)37 b(Morton,)j(Algebraic)c (dynamics)g(of)h(p)s(olynomial)e(maps)h(on)h(the)g(algebraic)163 1790 y(closure)30 b(of)g(a)h(\014nite)e(\014eld)g(I)s(I,)h Fb(Ro)s(c)m(ky)h(Moun)m(tain)f(J.)g(of)h(Math.)g Fm(24)g Fq(\(1994\))i(905{932.)21 1978 y([5])47 b(S.)26 b(D.)i(Cohen)e(and)g (D.)i(Hac)m(hen)m(b)s(erger,)g(Actions)f(of)g(linearized)e(p)s (olynomials)f(on)i(the)h(algebraic)163 2091 y(closure)j(of)g(a)h (\014nite)e(\014eld,)g Fb(Con)m(temp)s(orary)h(Mathematics)p Fq(,)i Fm(225)f Fq(\(1999\))i(17{32.)21 2278 y([6])47 b(S.)39 b(D.)g(Cohen)g(and)f(D.)i(Hac)m(hen)m(b)s(erger,)j(The)38 b(dynamics)g(of)h(linearized)e(p)s(olynomials,)i Fb(Pro)s(c.)163 2391 y(Edim)m(b.)29 b(Math.)i(So)s(c.)p Fq(,)g Fm(43)f Fq(\(2000\))j(113{128.)21 2579 y([7])47 b(W.)29 b(F)-8 b(on)m(tana,)31 b(Algorithmic)d(Chemistry)-8 b(,)28 b(in)f Fb(Arti\014cial)g(Life)h(I)s(I,)h Fq(SFI)f(Studied)f(in)g(the)i (Sciences)163 2692 y(of)h(Complexit)m(y)-8 b(,)30 b(v)m(ol)h(X,)g (\(1990\))h(150{209.)21 2880 y([8])47 b(W.)29 b(F)-8 b(on)m(tana)30 b(and)e(L)g(W)h(Buss,)f(\\The)h(arriv)-5 b(al)27 b(of)h(the)h(\014ttest",)h(to)m(w)m(ard)f(a)g(theory)g(of)f (biological)163 2992 y(organization,)i Fb(Bulletin)f(of)h(Mathematical) h(Biology)g Fm(56)g Fq(\(1994\))i(1{64.)21 3180 y([9])47 b(H.)30 b(Hasse,)i Fb(Num)m(b)s(er)d(theory)p Fq(,)i(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork,)31 b(\(2000\).)-24 3368 y([10])47 b(N.)28 b(Kataok)-5 b(a)29 b(and)e(K.)g(Kanek)m(o,)j(F)-8 b(unctional)27 b(Dynamics)g(I:)g(Articulation)f(pro)s(cess,)i Fb(Ph)m(ysica)f(D)p Fq(,)163 3481 y Fm(138)k Fq(\(2000\))i(225{250.)-24 3668 y([11])47 b(N.)41 b(Kataok)-5 b(a)43 b(and)d(K.)h(Kanek)m(o,)k(F) -8 b(unctional)41 b(Dynamics)f(I)s(I:)h(Syn)m(tactic)g(structure)g (preprin)m(t)163 3781 y Fa(http://arXiv.org/abs/ad)o(ap-o)o(rg/9)o(907) o(005)24 b Fq(\(2001\))33 b(\(subm.)c(to)i Fb(Ph)m(ysica)g(D)p Fq(\).)-24 3969 y([12])47 b(N.)30 b(Koblitz,)g Fb(Algebraic)g(asp)s (ects)g(of)h(cryptograph)m(y)p Fq(,)g(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork,)31 b(\(1997\).)-24 4156 y([13])47 b(R.)33 b(Lidl)e(and)i(H.)h(Niederreiter,)f Fb(Finite)g(\014elds)p Fq(,)g(Encyclop)s(edia)e(of)j(Math.)g(and)f(its)g(Appl.,)g(v)m(ol.)163 4269 y(20,)e(Addison-W)-8 b(esley)g(,)31 b(Reading,)f(Mass.)h (\(1983\).)-24 4457 y([14])47 b(D.)31 b(A.)f(Marcus,)h Fb(Num)m(b)s(er)e(\014elds)p Fq(,)h(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork)31 b(\(1977\).)-24 4645 y([15])47 b(P)-8 b(.)31 b(J.)f(McCarth)m(y,)h(Algebraic)f(extensions)g(of)g (\014elds,)g(Do)m(v)m(er,)i(New)e(Y)-8 b(ork,)32 b(\(1966\).)-24 4832 y([16])47 b(P)-8 b(.)30 b(Morton,)g(On)e(certain)i(algebraic)f (curv)m(es)g(related)g(to)h(p)s(olynomial)d(maps,)i Fb(Comp)s(os.)g (Math.)p Fq(,)163 4945 y Fm(103)p Fq(,)i(\(1996\))i(319{350.)1702 5225 y(21)p eop %%Page: 22 22 22 21 bop -24 573 a Fq([17])47 b(P)-8 b(.)37 b(Morton)h(and)f(P)-8 b(.)37 b(P)m(atel,)j(The)d(Galois)f(theory)i(of)f(p)s(erio)s(dic)e(p)s (oin)m(ts)h(of)h(p)s(olynomial)d(maps,)163 686 y Fb(Pro)s(c.)c(London)g (Math.)h(So)s(c.)f Fm(68)h Fq(\(1994\))i(225{263.)-24 873 y([18])47 b(P)-8 b(.)31 b(Morton)h(and)f(J.)g(H.)g(Silv)m(erman,)f (P)m(erio)s(dic)g(p)s(oin)m(ts,)h(m)m(ultiplicities,)c(and)k(dynamical) e(units,)163 986 y Fb(J.)h(fuer)g(reine)f(und)g(angew.)i(Math,)g Fm(461)g Fq(\(1995\))i(81{122.)-24 1174 y([19])47 b(P)-8 b(.)25 b(Morton)g(and)f(F.)i(Viv)-5 b(aldi,)23 b(Bifurcations)h(and)g (discriminan)m(ts)e(for)i(p)s(olynomials)e(maps,)k Fb(Non-)163 1287 y(linearit)m(y)j Fm(8)h Fq(\(1995\))j(571{584.)-24 1474 y([20])47 b(R.)30 b(W.)h(K.)f(Odoni,)e(The)i(Galois)g(theory)g(of) g(iterates)h(and)e(comp)s(osites)h(of)g(p)s(olynomials,)e Fb(Pro)s(c.)163 1587 y(London)h(Math.)i(So)s(c.)p Fq(,)g Fm(51)p Fq(,)g(\(1985\),)i(385{414.)-24 1775 y([21])47 b(F.)31 b(Viv)-5 b(aldi,)28 b(Dynamics)i(o)m(v)m(er)i(irreducible)27 b(p)s(olynomials,)g Fb(Nonlinearit)m(y)i Fm(5)i Fq(\(1992\))i(941{960.) -24 1963 y([22])47 b(F.)28 b(Viv)-5 b(aldi)25 b(and)i(S.)g(Hatjisp)m (yros,)h(Galois)f(theory)h(of)g(p)s(erio)s(dic)d(orbits)h(of)i (rational)f(maps,)h Fb(Non-)163 2076 y(linearit)m(y)h Fm(5)h Fq(\(1992\))j(961{978.)-24 2263 y([23])47 b(B.)31 b(L.)f(v)-5 b(an)30 b(der)g(W)-8 b(aerden,)31 b Fb(Algebra)p Fq(,)g(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork)31 b(\(1991\).)1702 5225 y(22)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF