Math 332

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
- radical ideal
- 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.