Math 332: Abstract Algebra

## Review for Final Exam

The final exam will cover the following material. For the definitions, you should know the statements, examples, and non-examples. For the theorems, there are two sections. The first just requires statements of theorems (including all hypotheses). The second requires full statements and proofs.

### Groups

• Definitions
• group (1.1, p. 2)
• permutation group, the symmetric group of degree n, cyclic group, dihedral group, alternating group, general linear group, special linear group, free group
• the parity of a permutation
• subgroup (p. 7), index of a subgroup
• order of a group, order of an element
• k-cycle (p. 16)
• group homomorphism (13.1, p.20), isomorphism (14.1, p. 21), automorphism (14.16, p. 23)
• the kernel and image of a group homomorphism (13.5 p. 20)
• conjugates of an element, conjugates of a subgroup, conjugacy classes of elements and of subgroups
• group action (15.1, p.24)
• conjugation action by a subgroup (15.8, p. 24)
• stabilizer, orbit (15.11, p. 25), centralizer (19.3, p. 30)
• left and right cosets (16.1, p. 26)
• class equation (19.5, p. 31---know what it means)
• external direct product (20.1, p. 32)
• normal subgroup (21.1, p. 35), normalizer of a subgroup (p. 35)
• factor/quotient group (22.1, p. 36)
• simple group (Tuesday, week 6)
• external (20.1, p. 32) and internal (23.4, p. 38) direct products/sums
• exact sequence, short exact sequence (Thursday lecture, week 6)
• semi-direct product, splitting of an exact sequence (as given in the Thursday lecture, week 7)
• composition series, factors of a composition series (Thursday lecture, week 7)
• elementary divisors, rank, and torsion part of a finitely generated abelian group (Tuesday lecture, week 7)
• p-subgroup, Sylow p-subgroup (Wikipedia)
• Statements of theorems
• subgroup tests (Tuesday lecture, week 2)
• the main results for cyclic groups outlined on p. 4 of the Thursday lecture, week 2
• the division algorithm and Euclidean algorithm for the integers (Thursday lecture, week 4)
• the orbit-stabilizer theorem (Tuesday lecture, week 5)
• Burnside's lemma (Tuesday lecture, week 5, and 18.4, p. 29)
• Chinese remainder theorem (Thursday lecture, week 5).
• Jordan-Hölder theorem (Thursday lecture, week 7)
• Structure theorem for finitely generated abelian groups (Tuesday, week 7) (Recall the Smith normal form of a matrix.)
• Sylow theorems (Wikipedia)
• Statements and proofs of theorems
• Cayley's theorem (15.6, p. 24)
• Lagrange's theorem (16.2, 16.3, 16.4, p. 26---know all the steps)
• Basic corollaries of Lagrange's theorem (16.5, 16.6, 16.7, 16.10)
• first isomorphism theorem (24.1, p. 39) and the resulting characterization of normal subgroups (24.3 p. 39)
• Other
• Know how to use Burnside's lemma to count modulo symmetry (Tuesday lecture, week 5).

### Rings

• Definitions
• ring (27.1, p. 46), with identity, commutative (27.3, p. 46)
• subring (Tuesday lecture, week 8)
• homomorphism (Tuesday lecture, week 8)
• kernel (29.2, p. 48)
• left ideal, right ideal, ideal (29.1, p. 48)
• characteristic (30.1, p. 50)
• factor/quotient ring (31.1, p. 50)
• zero-divisors (32.1, p. 51)
• integral domain (32.3, p. 51)
• prime ideal (33.1, p. 52),
• maximal ideal (33.8, p. 53)
• pid (34.4, p. 55, Thursday lecture, week 10)
• irreducible element (Tuesday lecture, week 11)
• monomial ordering (Tuesday lecture, week 12)
• initial ideal (Tuesday lecture, week 12)
• Gröbner basis (Tuesday lecture, week 12)
• Statements of theorems
• subring test (Tuesday lecture, week 8)
• the division algorithm in one variable (Thursday lecture, week 10)
• Gauss' lemma (35.7, p.57, Thursday lecture, week 11)
• Eisenstein's criterion (35.9, p. 58, Thursday lecture, week 11)
• Statements and proofs of theorems
• first isomorphism theorem and resulting characterization of ideals (Tuesday lecture, week 8)
• A finite integral domain is a field (32.8, p. 51)
• In a commutative ring R with 1, and ideal I is prime iff R/I is a domain. (33.7, p. 53)
• In a commutative ring R with 1, and ideal I is maximal iff R/I is a field. (33.12, p. 53)
• Every maximal ideal in a commutative ring with 1 is prime. (33.13, p. 54)
• If k is a field, then k[x] is a pid. (34.5, p.. 55, Tuesday lecture, week 11)
• If k is a field, f is in k[x], and a is in k, then f(a)=0 iff x-a divides f. (Tuesday lecture, week 11)
• If k is a field, and f is a nonzero element of k[x], then f has a finite number of zeros.
• In a pid, a nonzero ideal is prime iff it is maximal. (Thursday lecture, week 11)
• Other
• Know how to use the Buchberger algorithm to calculate a Gröbner basis. (Tuesday lecture, week 12)

### Algebraic Geometry

The PCMI handouts and class lectures are our references for all of the following.
• Definitions
• zero sets, algebraic sets, Z(I)
• the ideal of a subset of affine space, A(X)
• irreducible algebraic set
• Noetherian ring
• the affine coordinate ring of an algebraic set
• nilpotent element of a ring
• reduced ring
• finitely generated k-algebra
• morphism of algebraic sets
• Statements of theorems
• the Nullstellensatz
• Hilbert basis theorem
• Over an algebraically closed field, the categories of affine k-algebras and algebraic sets are equivalent.
• Statements and proofs of theorems
• For ideals I, J, if I is contained in J, then Z(I) contains Z(J).
• An algebraic set is irreducible iff its ideal is prime.
• Let k be algebraically closed. A ring is a finitely generated k-algebra iff it is the coordinate ring for an algebraic set.
• Other
• Know how to go back and forth between mappings of k-algebras and mappings of algebraic sets.

### Fields

All references in this section are to the Tuesday lecture of week 13. For the following, let k be a subfield of K.
• Definitions
• algebraic element in an extension field
• the irreducible polynomial of an algebraic element
• the algebraic closure of k in K
• algebraic extension
• algebraically closed field
• Statements and proofs of theorems
• If an element a in K is algebraic over k, then k[a] = k(a).
• An element a in K is algebraic over k iff [k(a):k] is finite.