MATH 401, FINAL EXAM

DANIEL R. GRAYSON

Problem 1 (25 points). True or false? (No support needed for your answer.) (a) ℤ/12 ⊕ ℤ/15 is a cyclic group.

(b) 𝔽_{243}^{×} is a cyclic group.

(c) 𝔽_{2}^{×} is a free abelian group.

(d) As a ℤ-module, ℂ^{×} is torsion-free.

(e) In a commutative ring, any maximal ideal is prime.

(f) Any finitely generated ℚ-module is free.

(g) Any torsion-free ℤ-module is free.

Problem 2 (15 points). Produce an explicit isomorphism between the general linear group Gl_{2} 𝔽_{2} and the symmetric group S_{3}.

Problem 3 (15 points). Suppose a finite group G acts transitively on sets X and Y, and there is a function f : X → Y compatible with the action in the sense that f(σx) = σf(x) for all σ ∈ G and all x ∈ X. Show all the fibers of f have the same cardinality.

Problem 4 (15 points). What is the remainder of 1999^{1999^{1999}} upon division by 11?

Problem 5 (15 points). Suppose I and J are ideals of a commutative ring R. Show that the subsets Z(IJ) and Z(I) ∪ Z(J) of Spec R are equal.

Problem 6 (25 points). Let ζ = e^{2πi/100} ∈ ℂ, and let K = ℚ(ζ). Let f(T) be the irreducible polynomial of ζ over ℚ. Suppose η ∈ ℂ is any other root of f(T).

(a) Show that f(T) | T^{100} − 1.

(b) Show there is an isomorphism θ : K → ℚ(η) with θ(ζ) = η.

(c) Show that η = ζ^{m} for some integer m.

(d) Show that K = ℚ(η).

(e) Show that gcd(m, 100) = 1.

(f) Show that |(ℤ/100)^{×}| = 40.

(g) Show that deg f ≤ 40.

(h) Show that [K : ℚ] ≤ 40.

Problem 7 (15 points). Suppose p is a prime number. Show that (ℤ/pℤ)^{×} is a quotient group of (ℤ/p^{2}ℤ)^{×}. Show that (ℤ/p^{2}ℤ)^{×} is a cyclic group.

Problem 8 (15 points). Suppose that R is a Euclidean domain.

(a) Describe the steps of the algorithm for diagonalizing a matrix A : R^{m} → R^{n} with row and column operations.

(b) Take R = ℤ and diagonalize the matrix B = (\begin{matrix}4 & −2 & −2 \\ 0 & 12 & 6 \\ −4 & 14 & 8 \end{matrix}).

Date: December 14, 1999.

(c) Identify the ℤ-module M = coker B up to isomorphism. What is its rank? What is the order of its torsion subgroup Tor(M)?

Problem 9 (15 points). Suppose R is a ring, M is an R-module, and N is a submodule. Show that if N and M/N are noetherian modules, then so is M.

Problem 10 (15 points). How many conjugacy classes of elements of order 5 are there in Gl_{3} ℚ? … in Gl_{4} ℚ? … in Gl_{3} ℂ?

Problem 11 (15 points). Suppose 0 → M →^{f} N →^{s} P → 0 is an exact sequence of R-modules, and f has a left inverse g : N → M, i.e., that g ∘ f = 1_{M}. Show that

the map N \underrightarrow{(\begin{matrix}g \\ s \end{matrix})} M ⊕ P is an isomorphism of R-modules.

Problem 12 (15 points). Compute the Jordan normal form of (\begin{matrix}3 & −1 & 1 \\ 2 & 0 & 1 \\ −2 & 1 & 0 \end{matrix}).

University of Illinois at Urbana-Champaign

   E-mail address: dan@math.uiuc.edu
   URL: http://www.math.uiuc.edu/~dan