Problem 1 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Let $K = Q(\sqrt{-3})$, an imaginary quadratic field.

(a) Does there exists a finite Galois extension $L/Q$ which contains $K$ such that $\mathrm{Gal}(L/Q) \cong S_3$? (Here $S_3$ is the symmetric group in 3 letters.)

(b) Does there exists a finite Galois extension $L/Q$ which contains $K$ such that $\mathrm{Gal}(L/Q) \cong Z/4Z$?

(c) Does there exists a finite Galois extension $L/Q$ which contains $K$ such that $\mathrm{Gal}(L/Q) \cong Q$? Here $Q$ is the quaternion group with 8 elements $\{\pm 1, \pm i, \pm j, \pm k\}$, a finite subgroup of the group of units $H^{\times}$ of the ring $H$ of all Hamiltonian quaternions.

Problem 2 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Let $f$ be a two-dimensional (complex) representation of a finite group $G$ such that $1$ is an eigenvalue of $f(\sigma)$ for every $\sigma \in G$. Prove that $f$ is a direct sum of two one-dimensional representations of $G$.

Problem 3 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Let $F \subset R$ be the subset of all real numbers that are roots of monic polynomials $f(X) \in Q[X]$.

(1) Show that $F$ is a field.

(2) Show that the only field automorphisms of $F$ are the identity automorphism $\alpha(x) = x$ for all $x \in F$.

Problem 4 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Let $V$ be a finite-dimensional vector space over $R$ and $T : V \to V$ be a linear transformation such that

(1) the minimal polynomial of $T$ is irreducible;

(2) there exists a vector $v \in V$ such that $\{T^i v \mid i \ge 0\}$ spans $V$.

Show that $V$ contains no non-trivial proper $T$-invariant subspace.

Problem 5 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Given a commutative diagram

$$
\begin{array}{ccccccccc}
A & \to & B & \to & C & \to & D & \to & E \\
\downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow \\
A' & \to & B' & \to & C' & \to & D' & \to & E'
\end{array}
$$

of Abelian groups, such that (i) both rows are exact sequences and (ii) every vertical map, except the middle one, is an isomorphism. Show that the middle map $C \to C'$ is also an isomorphism.

Problem 6 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Prove that a group of order $150$ is not simple.