Math 5530: Algebra Assignment 10
Spring 2011 Due Monday, 2 May Suppose that R is a PID and M is a finitely generated R-module such that M = Tor(M ). Show that there are cyclic submodules Z1 , . . . , Zr of M such that M Z1 Zr . =
Math 5530 Class Notes: 2/21/11
2.5. Algebraically closed eld and algebraic closure
A eld F is said to be algebraically closed if every polynomial in F [x] has a root in F . Since
a in F is a root of p(x) in F [x] if and only if x a divides p(x) (in F [x])
Math 5530 Class Notes: 2/16/11
2.2. Three observations and a proposition (cont.) Recall that L is a finite extension of F and G = AutF (L) = cfw_ : L L | is a ring homomorphism and |F = id . For a subgroup H of G, set LH = cfw_ L | () = G . Obviously F LG
Math 5530 Class Notes: 2/14/11
2. Fields and Galois Theory 2.1. Preliminaries Suppose that F and K are fields and : F K is a ring homomorphism. By convention (1F ) = 1K . Thus, ker = F ansd so since F is a field it must be that ker = cfw_0. It follows tha
Math 5530 Class Notes: 2/7/11
1.5. Localization (part 2) Recall that R is a commutative ring with identity, S is a multiplicative subset of R, and is the equivalence relation on R S given by (r, s) (r , s ) if there is a t in S such that t(rs - r s) = 0.
Math 5530 Class Notes: 1/31/11
1.4. EDs, PIDs, and UFDs (part 2, continued. . . )
Example 1.4.1 (Integers in quadratic number elds) Suppose that d is a square-free integer
and let Rd denote the set of algebraic integers in Q[ d].
It follows from general
Math 5530 Class Notes: 1/26/11
1.4. EDs, PIDs, and UFDs (part 2) We've seen that every Euclidean domain is a principal ideal domain. In this subsection we show that every principal ideal domain is a unique factorization domain. Recall that an integral dom
Math 5530 Class Notes: 1/24/11
1.2. EDs, PIDs, and UFDs (part 1) Suppose R is a commutative ring with identity and a and b are elements in R. (a) will denote the ideal generated by a. Thus (a) = Ra = cfw_ ra | r R . Define a relation | (divides) on R by a
Math 5530 Class Notes: 1/19/11
1. Commutative algebra and Galois theory In this section, unless otherwise stated, all rings are assumed to be commutative and to have an identity. 1.1. The division algorithm Proposition 1.1.1 (The division algorithm) Suppo
Math 5530 Class Notes: 3/2/11
3. More rings and ideals 3.1. Operations on Ideals Suppose that R is a ring and A and B are non-empty subsets of R. In contrast with the notation for subsets of a group, the notation AB does NOT denote the set of all products
Math 5530 Class Notes: 3/7/11
4. Modules Suppose k is a field, for example k = R or k = C. Recall that a k-vector space is a set, V , together with two operations, vector addition (a binary operation on V ) and scalar multiplication (a function k V V deno
Math 5530 Class Notes: 3/21/11
4.6. Submodules again An R-module, M is called simple or irreducible if M = cfw_ 0 and the only submodules of M are M and cfw_ 0 . Exercise 4.6.1. Suppose R is a ring with identity. (1) Show that if K is any left ideal in R
Math 5530: Algebra Assignment 8
Spring 2011 Due Wednesday, 13 April In these exercises, R,S, and T are commutative rings with identity, R S T , and 1R = 1S = 1T . Thus T is an extension of R and S and S is an extension of R. 1. (Transitivity in towers) (a
Math 5530: Algebra Assignment 7
Spring 2011 Due Monday, 4 April
1. Suppose R is a ring with identity, M is an R-module, and K M is a submodule of M . Show that M is Noetherian if and only if K and M/K are Noetherian. 2. Suppose that R is a commutative rin
Math 5530: Algebra Assignment 6
Spring 2011 Due Monday, 21 March
1. Suppose R is a ring, I is a two-sided ideal in R, and M is an R-module. Define IM = cfw_ r1 m1 + . . . rn mn | r1 , . . . , rn I, m1 , . . . , mn M . (a) Show that IM is a submodule of M
Math 5530: Algebra Assignment 5
Spring 2011
Due Monday, 7 March
There is no reason to turn in any problems you turned in with Assignment #4. For questions #2
and #3, you can (should?) use any parts of the Fundamental Theorem of Galois Theory you need,
or
Math 5530: Algebra Assignment 4
Spring 2011 Due Monday, 28 February
1. Show that a finite extension L of F is a Galois extension if and only if |AutF (L)| = |L : F |. 2. Suppose that L is the splitting field of x4 - 2 (in Q C). Find |L : Q|, AutQ (L), and
Math 5530: Algebra Assignment 3
Spring 2011 In these exercises, F , K, and L are fields. 1. Suppose that K is an extension of F . Show that if K/F is finite, then K/F is algebraic. 2. Suppose that F , K, and L are fields with F K L. (a) Show that L/F is a
Math 5530: Algebra Assignment 2
Spring 2011 Due Monday, 14 February
1. Suppose that R is a commutative ring with identity and S is a multiplicative subset of R. r Let : R S -1 R by (r) = 1 . (a) Show that if J is an ideal in S -1 R, then S -1 ( -1 (J) = J
Math 5530: Algebra Assignment 1
Spring 2011 Due Monday, 31 January
1. Show that every ED is a PID. 2. Suppose that R is a PID and r in R is a non-zero, non-unit. Show that TFAE: (a) r is prime. (b) r is irreducible. (c) (r) is a maximal ideal. 3. Suppose
Math 5530 Class Notes: 5/2/11
6. Structure of Modules (cont.) 6.3. Completely reducible modules Motivation. Suppose k is a field, V is a non-zero k-vector space, and cfw_ v | A is a basis of V . For in A, let L = kv be the one-dimensional subspace of V s
Math 5530 Class Notes: 4/11/11
5. More Commutative Rings (cont.) 5.5. Application: Invariant Theory of Finite Groups (cont.) Exercise 5.5.7. Suppose that R is a Noetherian ring. Show that R[x1 , . . . , xn ] is a Noetherian ring. Recall that k is a field,
Math 5530 Class Notes: 4/6/11
5. More Commutative Rings (cont.)
5.4. Finitely Generated R-algebras
In this subsection, all rings are assumed to be commutative.
Finitely generated R-modules. Suppose that R is a commutative ring with identity. An
R-module M
Math 5530 Class Notes: 4/4/11
5. More Commutative Rings (cont.) 5.2. R-algebras and R-algebra Homomorphisms Unlike the definition of a group, ring, or module, the definition of an algebra may vary slightly from author to author. However, in the case of al