General Exam in Algebra

May, 2003

Problem 1

Let A∈Mn(ℝ)A \in M_{n}(\mathbb{R}) be an alternating matrix, i.e. tA=−A{ }^{t} A=-A. Show that if nn is odd then det⁡A=0\operatorname{det} A=0.

Problem 2

Let A,B∈Mn(ℂ)A, B \in M_{n}(\mathbb{C}) be two commuting matrices. Prove that they have a common eigenvector in ℂn\mathbb{C}^{n}, i.e. there exists a nonzero v∈ℂnv \in \mathbb{C}^{n} such that Av=λvA v=\lambda v and Bv=μvB v=\mu v for some λ,μ∈ℂ\lambda, \mu \in \mathbb{C}.

Problem 3

Let U={(1a12a1301a23001):aij∈ℝ}U=\left\{\left(\begin{array}{ccc}1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1\end{array}\right): a_{i j} \in \mathbb{R}\right\}. Prove that UU is a group for multiplication of matrices and identify the center Z(G)Z(G) and the commutator subgroup [G,G][G, G] (we recall that [G,G][G, G] is the subgroup generated by all commutators xyx−1y−1x y x^{-1} y^{-1} with x,y∈Gx, y \in G.

Problem 4

Show that the group G1G_{1} of all real numbers for addition is isomorphic to the group G2G_{2} of all positive real numbers for multiplication. Furthermore, show that the group H1H_{1} of all rational numbers is not isomorphic to the group H2H_{2} of all positive rational numbers for multiplication.

Problem 5

Let GG be a group such that there exists a surjective group homomorphism G→ℤG \rightarrow \mathbb{Z}. Prove that for any subgroup of finite index H⊂GH \subset G there also exists a surjective group homomorphism H→ℤH \rightarrow \mathbb{Z}.

Problem 6

Let AA be the ring of all continuous real-valued functions on [0,1][0,1]. Give an example of a maximal ideal in AA. Furthermore, give an example of an element f∈Af \in A which is not invertible, but which is not a zero divisor either.

Problem 7

Let f(x)∈ℝ[x]f(x) \in \mathbb{R}[x] be a nonzero polynomial. Show that there exists a number r∈ℝr \in \mathbb{R} such that the polynomials f(x)f(x) and f(x+r)f(x+r) are relatively prime.

Problem 8

Let K=ℤ/pℤK=\mathbb{Z} / p \mathbb{Z} be the field of pp elements (where pp is a prime). For an integer d>0d>0, we let

σd=∑x∈Kxd\sigma_{d}=\sum_{x \in K} x^{d}

Show that

σd={−1 if (p−1) divides d0 otherwise \sigma_{d}=\left\{\begin{aligned} -1 & \text { if }(p-1) \text { divides } d \\ 0 & \text { otherwise } \end{aligned}\right.