HOMEWORK FOR MODERN ALGEBRA STUDENTS

Math 6140: Modern Algebra 2, Spring 2008

Homework

Assignment	Assigned	Due	Problems
HW1	1/16/08	1/23/08	Read: Sections 10.1 and 10.2. You should be able to solve 10.1.1-7 at a glance. Problems 10.1.8-10 introduce definitions that you should know and facts you should be able to prove. Problems 10.1.18-20 provide important examples, which you should understand. You should be able to solve 10.2.1-4 at a glance. Turn in: 1. Let A be the 2-element semigroup of integers under addition modulo 2, let B be the 2-element semigroup of integers under multiplication modulo 2, and let S be the product of these semigroups. (a) Write the operation table for the binary operation of S. (b) Find the congruences of S. (c) Draw the congruence lattice of S. 2. Show that no group has a congruence lattice isomorphic to the lattice in 1(c). 3. Let G be the group of integers, and let A be the G-set with two 2-element orbits. (a) Find the congruences of A. (b) Draw the congruence lattice of A. (c) Show that no group has a congruence lattice isomorphic to the lattice in 2(b).
HW2	1/23/08	1/30/08	Read Section 10.3. You should be able to solve 10.2.6, 10.3.1, 10.3.6 at a glance, and these results are worth remembering. Turn in: 1. State and prove the version of Exercise 10.2.11 that holds for algebras in any language. (To do this right you will have to define ``product congruence''.) 2. Do Exercise 10.3.7. Is the corresponding statement true for algebras in any language? 3. Exercise 10.3.11.
HW3	1/30/08	2/6/08	Read Appendix II Turn in: 1. (a) Show that a subset of an independent set is independent. (b) Show that if C is closed under the formation of subalgebras, F is free over X relative to C, and Y is a subset of X, then the subalgebra of F generated by Y is free over Y relative to C. 2. Read Exercise 10.3.23, then do Exercise 10.3.24. 3. Exercise 10.3.27.
HW4	2/6/08	No due date	Read Section 10.4 and parts of 10.5 (Definitions of exact, split exact, projective, injective and flat.) You should be able to solve 10.4.1-7 and 14 at a glance. Exercise 10.4.8 is useful to know when proving that the field of fractions of an integral domain is flat. (But note that the definition of the equivalence relation is not, in fact, an equivalence relation. The correct definition is: (u,n) is equivalent to (u',n') iff u''(u'n-un')=0 for some u'' in U.) Make sure you understand how to do 10.4.11-12. The facts stated in 10.4.14-15 are worth remembering. (To do 10.4.15, use 10.4.8(d).) 1. Let R be a commutative ring. Show that if M and N are finitely generated (cyclic) R-modules, then so is the tensor product of M and N over R. Now suppose that R, S and T are noncommutative rings, that M is f.g. (cyclic) as an (R,S)-bimodule, N is f.g. (cyclic) as an (S,T)-bimodule. Must the tensor product of M and N over S be f.g. (cyclic) as an (R,T)-bimodule? 2. Use the result of Example 4, pg 369, to show that no unital ring has additive group isomorphic to Q/Z. 3. Show that the tensor product of two commutative rings is their coproduct in the variety of commutative rings.
HW5	2/13/08	No due date	Much of the material in Section 10.5 can be safely set aside until you study homological algebra more thoroughly. Still, projective and injective modules come up in many contexts, so you should familiarize yourself with the results of Exercises 10.5.3-13. (Note: direct sum should be product in 10.5.4.) 1. Explain why it is possible but not reasonable in module theory to define the tensor product of infinitely many modules. 2. Let Ab be the variety of abelian groups. Show that the nth-power functor F(A) = A^n from Ab to Ab is both exact and representable. What is the representing object? 3. Describe the representable functors from the variety of sets to the variety of groups. (You must determine the cogroup cooperations on an arbitrary set.)
HW6	2/20/08	2/27/08 (Postponed until 2/29/08)	Read Sections 11.1-3. You should be able to solve all exercises in Section 1 at a glance, except possibly for establishing the ``strictly larger'' clause in Exercise 11.1.14. The exercises in Section 11.2 that involve finite dimensional spaces but not the tensor product are really undergraduate linear algebra problems, so make sure that you can do them. Try Exercise 11.2.9, which involves infinite dimensional spaces. Find a necessary and sufficient condition for an endomorphism of a space to induce a kernel-image decomposition, generalizing Exercise 11.2.11. Turn in: 1. Consider the complex numbers to be a 2-dimensional vector space over the real numbers, with ordered basis (1,i). (a) If r = a+bi, what is the matrix for the transformation L_r(x) = rx (left multiplication by r)? (b) What are the trace and determinant of [L_r]? (c) Repeat this exercise with the rational numbers in place of the real numbers and Q[cube_root(2)] in place of the complex numbers. 2. A bilinear form on an F-vector space V is a bilinear function b(x,y): V x V -> F. The form is symmetric if b(x,y)=b(y,x) for all x, y, alternating if b(x,x)=0 for all x, degenerate on the left if there is some nonzero vector v such that b(v,x)=0 for all x, and degenerate on the right if b(x,v)=0 under the same conditions, and nondegenerate if it is not degenerate on either side. (a) If B=(w1,...,wn) is an ordered basis for V, define the matrix of the form b relative to B to be the matrix [b] whose ij-th entry is b(wi,wj). Show that, when written in coordinates relative to B, b(u,v) = [u]^t * [b] * [v]. (b) When R = reals and b(x,y) = dot product on R^n, what is the matrix for b relative to the standard basis? Some other basis? (c) Describe what it means for the matrix of b if b is symmetric, alternating, degenerate on one side, or nondegenerate. 3. Let F be a field. (a) Show that the elements (f^i)_{i<omega}, f in F, are F-linearly independent elements of F^{omega} (b) Compute the F-dimension of F^k for any cardinal k. (c) If V is an F-space, what is the dimension of the dual space, V^*?
HW7	2/27/08	3/5/08	Read Section 11.4, taking special note of Cor 27, Thm 29, and Thm 30, which were not proved in the lecture. Cor 27 explains how to interpret nonzero determinant over integral domains (it does not mean invertibility), while Thm 30 shows how to test for invertibility with the determinant. Turn in: 1. 11.4.6 2. Let S is a finite subset of the unit interval that includes both endpoints. Assume that every element in S other than 0, 1 is the average of two other elements of S. Show that every element of S is rational. (Hint: Use a slight refinement of Minkowski's Criterion.) Solution 3. Show that if an algebra satisfies DCC on subalgebras, then every 1-1 endomorphism is an automorphism. Similarly, show that if it satisfies ACC on congruences, then every surjective endomorphism is an automorphism.
HW8	3/5/08	3/12/08	Read Sections 12.1-2, possibly also 12.3 (although may not get that far this week). Here is an exercise to ponder, but not turn in. Let A be an mxn matrix over a commutative ring R. The k-th determinantal ideal of A, D_k(A), is the ideal of R generated by all determinants of kxk minors of A. Show that if matrices A and B are equivalent (meaning that B = PAQ for invertible P and Q), then D_k(A)=D_k(B) for all k. Use this to explain how to compute the invariant factors of a f.g. module over a PID using determinants. Turn in: 1. 12.1.2 2. Let R be an integral domain and Q be its field of fractions. (a) Show that the functor of tensoring with Q is an exact functor from R-modules to Q-vector spaces. (Reread 10.4.8, and note the comment on this exercise above in HW4.) (b) Show that the Q-dimension of (Q tensor M) equals the rank of M. (Ex. 12.1.20.) 3. Let M be the module presented by generators G = {x, y, z} and relations R = {2(1+i)x - 3(1-i)y + (1+i)z = 0, 4(1-i)x + 7(1+i)y + (1-i)z = 0, 6ix - 10y + 2iz = 0} over the PID Z[i]. (a) Write M in invariant factor form. (b) Express the new generators in terms of the original generators. Notes Solutions
HW9	4/2/08	4/9/08	Read 13.1-4. Exercises 13.1-3 are about calculating in quotient rings, which you should make sure you know how to do. Exercise 5 proves that the rational algebraic integers are exactly the integers, which is worth remembering. Exercises 13.2.1-14 should appear routine to you. The other exercises in 13.2 are harder, but all are important. Section 13.3 is not directly in main thread of the chapter, but you should be able to determine, say, which angles of the form 2*Pi/n are constructible. 1. Given a triangle ABC, the length of the angle bisector of the angle at A is the length of the segment from A to the side BC along the angle bisector. It is easy to see that if a triangle is constructible, then the lengths of its angle bisectors are constructible numbers. Is the converse true? (That is, given three constructible lengths that are the lengths of the angle bisectors of some triangle, are they the lengths of the angle bisectors of a constructible triangle?) 2. The finite field F_q with q=p^n elements is the splitting field for x^q-x. (a) Show that the lattice of subfields of F_q is isomorphic to the lattice of divisors of n. (b) Show that every finite field is a simple extension of its prime field. 3. Show that any algebraically closed field is infinite.
HW10	4/9/08	4/16/08	Read Section 13.5-6. You should be able to solve 13.5.1-8. Exercises 13.6.1-6 should all be easy. Exercise 13.6.13 and the sequence 13.6.14-17 both establish nontrivial theorems. You should know the theorems even if you don't solve the problems. 1. 13.5.5 2. 13.6.11 3. Prove or disprove: a coefficient of a cyclotomic polynomial is either 0, +1, or -1.
HW11	4/16/08	4/23/08	Read Sections 14.1-3. The exercises in 14.1 should all look easy. Make sure you know how to do 14.1.4! The exercises in 14.2 are computational, but you need to be able to do those computations. Try 14.2.1-8 for practice. Try 14.3.1-5 and 14.3.10 also. 1. 14.1.7 2. 14.2.4 3. 14.2.23 (read 14.2.17 first).
HW12	4/23/08	4/30/08	Read Sections 14.4,4-5-9. We won't have time to say much about them in the lecture. Theorem 14.4.25, the statement of the Kronecker-Weber Theorem, and the statement of Krull's Theorem are especially important. 1. Prove that a complex number is constructible iff the Galois group of its minimal polynomial over Q is a 2-group. Show that there is a nonconstructible number whose minimal polynomial over Q has degree equal to a power of 2. 2. Show that the following are equivalent for complex numbers alpha and beta. (a) Every automorphism of C that fixes alpha also fixes beta. (b) There is a rational function r(x) in Q(x) such that r(alpha)=beta. 3. Let E/F be an extension without intermediate fields. Show that if the minimal polynomial of some element alpha in E has 2 distinct roots in E, then this polynomial splits over E.