University:
Boston University
Course:
GRS MA 742 | Algebra II
Academic year:

2023
Views:

181

Pages:

2
Author:

Mary Mueller

Preliminary Exam Afternoon Exam (3 hours) Part I Solve four of five problems. Problem 1. Let T : R4 → R3 be the linear map T (x1 , x2 , x3 , x4 ) = (x1 + 2x2 − x3 − x4 , 2x1 + 4x2 + x3 + 10x4 , x1 + 2x2 + x3 + 7x4 ) find a basis for the kernel of T . Problem 2. Show that the matrices A= 5 −6 1 0 and B = 7 −5 4 −2 are similar over R. In other words, show that there is an invertible matrix C with real coefficients such that A = C −1 BC. However, you do not need to exhibit C explicitly. Problem 3. Give an example of a polynomial p(x) ∈ Z[x] of degree 10 which is reducible modulo 2, 3 and 5 but irreducible over Z. Problem 4. Let V be the real vector space of bounded continuous functions f : R≥0 → R, and h∗, ∗i an inner product on V given by Z ∞ hf (x), g(x)i = f (x)g(x)e−x dx 0 Let W be the subspace of V spanned by f1 (x) = 1 and f2 (x) = e−x . Find an orthonormal basis for W . Problem 5. Let p(x) = x4 + 7x3 + 14x2 + 7x + 1 q(x) = x4 + 10x3 + 23x2 + 10x + 1 Find polynomials f (x), g(x) with rational coefficients such that f (x)p(x) + g(x)q(x) = 2x2 + 6x + 2 Part II Solve three out of six problems. Problem 6. Let Sn denote the symmetric group on n elements. Show that S12 contains an element of order 35 but no elements of order 33. What is the smallest n such that Sn contains an element of order 33 ? Problem 7. Let O(2, R) denote the group of orthogonal 2 × 2 matrices - i.e. 2 × 2 matrices A such that AAT = I, and GL(2, R) the group of invertible 2 × 2 matrices. Determine if O(2, R) is a normal subgroup of GL(2, R). Problem 8. Let T : R5 → R5 be a nonzero linear map such that the image of T is contained in the kernel of T . List the possibilities for the Jordan normal form of T. Be sure that your list is irredundant in the sense that no two matrices on your list are similar. Problem 9 Let V denote the vector space of polynomials in one variable with coefficients in R, and let T (f (x)) = xf (x). Prove that if W ⊂ V is a subspace such that T (W ) ⊂ W (i.e. W is stable under T ), then V /W is finite-dimensional. Problem 10 Let L ⊂ Z be the subgroup of Z3 generated by the elements (−1, −1, 4), (2, 4, 0) and (3, 3, 8). Write Z3 /L as a direct sum of cyclic groups. Problem 11 Suppose G and H are finite groups of relatively prime orders. Prove that Aut(G × H) is isomorphic to the direct product Aut(G) × Aut(H). Part III Solve one of three problems. Problem 12 (a) Show that for any n ∈ N, the ring Z has a chain of ideals {0} ( I1 ( I2 ( I3 · · · ( In ( Z. where ( denotes a strictly proper ideal (i.e. In 6= In+1 ) (b) Does Z have an infinite strictly increasing chain of ideals {0} ( I1 ( I2 ( I3 · · · ( In ( · · · Z.? If so, exhibit such a chain, and if not, give an example of a commutative ring R with such a chain. Problem 13 Define a sequence of fields Fn , F1 ⊂ F2 ⊂ F3 ⊂ · · · as follows. F1 = Q, and for n ≥S1, let Fn+1 be obtained from Fn by adjoining square roots of all elements of Fn . Let F = Fn . Prove that F does not contain a cube root of 2. Problem 14 Prove that the additive group of rational numbers Q is not finitely generated.

Algebra Preliminary Afternoon Exam

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