Algebra a graduate course

I Martin Isaacs Algebra A Graduate Course I Martin Isaacs Graduate Studies in Mathematics Volume 100 f I � , ,�American Mathematical Society - � :: "':': "' ,, Providence Rhode Island EDITORIAL COMMITTEE DaviJ Cox (Chair) Steven G Kralltz Rafe �Iazzeo �hlltill ScharlCl1lann 2000 Mllti!emlLtics Sllbjr.d C/Il.5sijirotion Primary OOAO;); SL'cuudary 12-01, l3�01, 16-01,20-01 For additional informatiun and \Jpdat� on thb; hook, visit www.ams.orgjbookpagesjgsm-lOO Library of Congress Cataloging_in_Publication Data \saacs, I /l.lartin, 1!l4{}- Algebra: a grilduilte course / I ?t.lllrtin bllacs p em - (GnHiuat.e studies in llmthematics , v 1(0) Originally publish d: Pacific Grove, Calif Drooks/Cole, c19!l4 Includ PI > > P", The height of every prime is finite, and, in fact, by Krull's theorem (27 19) we know that if P is a minimal prime over an m-generator ideal, then ht(P) ::: m We need a lemma (30.20) LEMMA Let R = F[XI X�J where F is a field If M is any maximal ideal of R, then ht(M) 2: n • By Corollary 27.26 the maximum possible height for any prime of the ring R is n, and so we can strengthen Lemma 30.20 and assert that all maximal ideals have height exactly equal to n For our present needs, the stated inequality is sufficient Proof of Lemma 30.20 We work by induction on n, noting that when n = 0, we have R = F and so the only maximal ideal is with height O Suppose rl > and let S = F[XJ, , X�_d � R Write Mo = M n S and note that Mo is an ideal of S Our first goal is to show that Mo is a maximal ideal Use overbars to denole the canonical homomorphism R � RIM = R,and note that R is a field Since F n M = 0, we have F ;:::;: F and we identify F with F Thus R = F[al a�J, where we have written aj to denote Xj By Zariski's theorem (30.8) we deduce that the elements aj are algebraic over F, and hence S = F[al , a�_d is a field Since S 3: SIMo, we conclude that Mo is a maximal ideal of S, as desired By the inductive hypothesis, the height of Mo in S is at least n I, and so we can find prime ideals Mj of S such that Mo > MI > > M�_I' We write P; = Mj [X�] � R to denote the sel of those polynomials in R = S[X�] that have all coefficients in the ideal Mj of S The Pi are clearly ideals of R that satisfy M ;2 Po > PI > > P�_I , • - and the proof will be complete if we can show that the Pi are prime ideals of R and that M > Po Since R = S[X,,] and Pi = Mi[X�J, it is clear thai RI Pj is isomorphic to the polynomial ring (SI Mj )[X"l overthe domain SI Mj • It follows that RI Pj is a domain and hence Pi is a prime ideal Furthennore RI Po is not a field and • hence Po is not maximal Thus Po < M, as required Proof of Theorem 30.18 Let I be the ideal of R generated by Q, and note thai I < R since Zer(l) = Zer(Q) is nonempty Suppose that P is a prime ideal minimal over I By Krull's theorem, we have ht(P) ::: I Q I < n, and so Lemma 30.20 guarantees that P is not a maximal ideal of R Now consider the algebraic set A = Zer(P) Since P is not a maximal ideal il follows by Corollary 30 J O that A is not a minimal algebraiC set, and 504 CHAPTER 30 so it contains more than one point Clearly, A is the union of the singleton subsets {a} for a E A, and each of these is an algebraic set by Theorem 30.15 P by Corollary 30.10, and since P is prime A is irreducible by Theorem 30 1 It follows that A is not a finite union of proper algebraic Now Pol(A) = sets We conclude that A contains infinitely many points and since Zer(Q) Zer( P) = Problems 30.1 A the result follows • _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Let E be algebraically closed and give En the Zariski topology (working over some field F � E) Prove that En is compact HINT: Show that a collection of closed sets with all finite intersections nonempty has a nonempty intersection 30.2 Suppose F � E where E is a finite field Show that the only irreducible algebraic sets over 30.3 Let F in En are the minimal nonempty algebraic sets R be a commutative ring that is finitely generated as an algebra over a field Show that the Jacobson radical of R is a nil ideal 30.4 £ S be commutative rings such that S = R[si S2 snl Let M be S and assume that M n R is maximal in R Show that M n T is maximal in T for every subring T of S such that T R Let R • • • a maximal ideal of 30.S Let E F be fields and assume that E is algebraically closed Let A £" En be maximal among proper irreducible algebraic subsets of En over F Show that A = Zer(!) where I is some prime element in F[X I , Xn l • 30.6 Suppose I(X, Y) and g(X, Y) are polynomials with coefficients in some algebraically closed field F , and suppose they have infinitely many common zeros in F2 Show that there is some nonconstant polynomial in that divides both F[X, Y] f and g HINT: If {f, g} is contained in infinitely many maximal ideals, show that this set is contained in some nonmaximal prime ideal Show that a nonmax­ imal, nonzero prime in 30.7 is necessarily principal R = F[XI , , Xn), where F is a field Let M be a maximal ideal of R, and suppose that I is any ideal Show that 1/1 M is finite dimensional as Let an 30.8 F[X, Yl F-vector space Let R = F[XI , prime ideal • Xn ], where F is a field and suppose that P a Show that there exists a field extension � R is L F and a point x E Ln that the transcendence degree tdF(L) S n and P = Pol(x) b Suppose E F, where E is algebraically closed and tdF(E) ?: n that every prime ideal of R has the fonn Pol(x) for some point x HINT: For part (a), observe that R/ P would have to a such Show E E" be embedded in L ALGEBRAIC SETS AND THE NULLSTELLENSATZ 30.9 Let A £ 505 E� be an algebraic set over F A point x E P is called a generic point for A if A = Zer(Pol(x)) (Note that necessarily, x E A in this case.) a Show that only irreducible algebraic sets can have generic points b If E is algebraic over F show that the algebraic sets with generic points are precisely the minimal nonempty algebraic sets c If every irreducible algebraic set in Eft has a generic point, show that either E is a finite field or the transcendence degree tdF(E) 2:: n d If E is algebraically closed and tddE) 2:: n , show that every irreducible algebraic set has a generic point HINT: For part (c), show by induction on Pol(Eft) = O n that if E is infinite, then Deduce that Eft is irreducible For part (d), use Problem 30.8 30.10 Let R = F[X I , , X,,] and suppose E F If every prime ideal of R has the form Pol(A) for some subset A £ En, show that Pol(Zer(l» = /J HINT: This problem can be done without appeal to the Nullstellensatz NOTE: Observe that if E is algebraically closed and tdF(E) :: n, then by Problem 30.8 the hypothesis is automatically satisfied In particular, this provides a comparatively easy proof of the Nullstellensatz in the case where F = Q and E = C 30.11 Let F be algebraically closed and suppose that A is anol necessarily associa­ tive algebra over F with dimF(A) x =n < 00 Assume that for every element E A, there exists a nonzero element y E A such that yx = O Show that there exists a nonzero element a E A such that ( ( (x)a)a) -)a = for all x E A, where there are n occurrences of a HINT: Use Theorem 30.19 in this fonnula Index Abel, N 274 abelian group 1 , 19, 37, 90-93, 139, 314 characters of 229 abelian X·group 142-153, 156, 157, 167 ACC (ascending chain condition) 143-145, 433 action of group 42-43 on cosets 44,45, 15 via automorphisms 95, 112 1 adjectives 163 adjoint matrix 439,456 affine group ofline 8, 29 afforded character 216 algebra (over a field) 166, 171, 172, 173, 213, 295 algebraic closure 267-268, 271 offinite degree 401-407,415 algebraic element 254-259,386, 398 algebraic field extension, definition of 256 algebraic geometry 493 algebraic integer 454, 463,469, 471-472 algebraic number 256 algebraic number field 256 algebraic numbers, field of 256, 259, 268, 410 algebraic point 494 algebraic set 493-505 algebraically closed field 267-269, 303, 355 algebraically independent elements 379-380, 383, 398 alternating group 75 simplicity of 77 annihilator (see also prime annihilator) 163, 176, 177, 210-2 1 , 440,442 antichain 157 archimedian ordered field Anin, E 145-146 461 Anin-Rees theorem 401 Anin-Schreier theorem Anin'5 theorem aninian module 412-413 260, 262, 389 170, 444 aninian ring (see right aninian ring or left artinian ring) aninian commutative ring 252, 428 145-146, 14S-149, 150, aninian X-group 157 ascending central series (see upper central series) ascending chain condition (see ACC) associated factors 338 440-442, 450 associated prime ideal associative propeny 173 augmentation ideal 173 augmentation map automorphism of group 19-22 of symmetric group automorphism group 79-80 of algebraically closed field of group 20, 28, 41 310 axiom of choice 405 5, 144-145, 153, 269 basis for free module 202, 434 Berlekamp algorithm bijection 333-339,464 binary operation 507 50S INDEX binomial coefficient 55 block diagonal representation 216, 219 bracket operation 305 Brodkey's theorem 66 Burnside, W 51 1 Burnside's fusion lemma 121 Burnside's theorem on nonnal p-complemenls 122 Burnside's two-prime theorern 66, 1 , 213, 467-471 cancellation in direct products 85, 139-140 canonical homomorphism 30, 32 cardinal number 269-270 casus irreducibilis 351 Cauchy-Frobenius theorem 50-5 Cauchy's theorem 54,57 Cayley, A Cayley's theorcm 12, 161 center of character 225, 229, 230 of dihedral group 19 of group 19, 39, 229 of p·group 63 of ring 174 center with finite index 1 9-120, 128 central chief factor 141 central extension 1 central product 97 central series 105-107 centralizer 18-19 in homomorphic image 53 in ring 174 centralizer ring of module 84 chain 143 character 213-230, 468-471 , 473 definition of 216 character induction and restriction 225-226 character table characteristic offield 214, 271 , 28 , 296-299, 308,326, 401, 404 of ring 173 characteristic polynomial 359-362 characteristic subgroup 19,21 chief factor 102, 141 chief series 102, 131 Chinese remainder theorem 334, 341, 431, 480 Classification theorem for simple groups 99 choice function 145 38, class (see conjugacy class) class equation 49 class function 217,221 226, 227 class number 480 classical adjoint 439,456 Clifford's theorem 158 closed elements in Galois connection 276 closureoperator 276 closure of cosel product 24 25 closure propeny 6,9, 14 coboundary 368 cochain 368 cocycle 368 coefficient of cyclotomic polynomial 309-310, 323 of polynomial 233, 272, 457 sequence 234 cohomology group 367-368, 369 colored cubes 53 colored squares 52-53 comaximal ideals 341,431 commutative diagram 32 commutative ring 160 commutator of elements 27, 37, 106, 1 of subgroups 27,37, 105-106, 110, 1 commutator subgroup (see derived subgroup) compass 315-316 complement in X-group 149 complement to nonnal subgroup 94, 377 completely reducible module 170, 197-198 completely reducible X-group 149-153, 158 complex conjugate 223 complex number field 264, 355, 316, 383 composition factor 38, 101, 102, 132 having specified type 140 of module 212 composition length 134, 146, 150-15 , 188 compositionof mappings 4,6,7, 12, 159 composition series 38, JOI, 102, 131 (seta/so X-composition series) compositum of fields 289, 2-313 conjugacy class 47 48, 120, 217, 468, 470 in symmetric group 73, conjugacy o f point stabilizers 77 conjugate elements 20, 120 conjugate pennutations 72-73 conjugate subgroups 20, 57, 77 conjugation action 43, 49 constant polynomial 235 constituent of representation 216, 219 constructible number 16-320, 325 construction of abelian groups 91 INDEX continuous functions, ring of 252, 429-430 contraclion 426-427 coprime elements of UFO coproduct 86 core 241 of subgroup 45, 13 of Sylow subgroup 66 Correspondence Iheorem 35, 36 137 141 of cyclic groups 90-93, 98, 251 of simple groups 96,97 direct sum (see direct product) of ideals (Set direct sum of rings) of rings 160, 199-200, 1 , 212, 334, 431, cosets, !lCtion on 44,45, 1 counting colored cubes 53 couming colored squares 52-53 counting conjugates 48-49 480 50 counting Sylow subgroups crossed homomorphism cycle 70, cycle structure cyclic group descending chain condition (Set DCC) detenninam 30, 165, 359-361 , 373 development of Sylow subgroups 57 diagonal matrix 223 diagonal subgroup 96,97 diamond 34, 40 Diamond theorem 33, 36 dihedral group 7, 12, 19, 21 , 47, 229 direct product 83-89, 107, 108, 134-135, for rings 164 for X-groups 130 coset 22-24 counting orbits 509 58-59 366-368, 370 72, 76, 465 16-18, 21 28, 91-92, 98, 260, 310 automorphismsof 28, 310,312 isomorphism of 18,32 cyclic Sylow subgroup 82, 118, 122, 127 cyclotomic field 310-313,315, 323-324 cyclotomic polynomial 246,308-3 1 , 314- 315, 323-324, 340 307 of X.simple X·groups 151-152 Dirichlel's theorem 314 discriminant of field extension 370-37 , 374, 463 of polynomial 371-376, 378, 398 disjoint cycle decomposition (see cycle structure) disjoint pennutations 70 distinct fOO{S 279, 295-296 divisible group 98 Division algorithm 236 division ring 165, 167, 168, 72, 184-185, 201 339 cyclotomy matrix ring over 169, 188-189, 191-192, 200 DCC (descending chain condition) onprime ideals 421, 443 Dedekind, R 474 143-145 Oedekind domain 436,464,474-492 Oedekind's lemma 27.40 on independence of field automorphisms 346, 366 degree of character 216,219, 468,473 offield extension 255-256,259, 289, 299 301, 302, 1 , 316, 35 , 354 of polynomial 236, 252.261 328, 351 ohepresentation 215 dense subring 185, 187 derivation 293-295, 304-3