Algebra I - January 2023

Algebra Qualifying Exam I (January 2023)

You have 120 minutes to complete this exam.

Problem 1.

(10 points) Let $G$ be a group of order 81. Is $G$ necessarily abelian? Please justify your conclusion.

Problem 2.

(10 points) Let $S_n$ be the symmetric group with $n\ge5$. Find all the normal subgroups of $S_n$. Please justify your conclusion.

Problem 3.

(10 points) Determine the greatest common divisor of the polynomials $f(x)=x^{100}-1$ and $g(x)=x^{17}+2x^6+x+100$ in the polynomial ring $\mathbb Q[x]$.

Problem 4.

(10 points) Consider the ring

$$\mathbb Z[\sqrt{-2}]=\{a+b\sqrt{-2}\mid a,b\in\mathbb Z\}.$$

Define $N(a+b\sqrt{-2})=a^2+2b^2$. Prove that $\mathbb Z[\sqrt{-2}]$ with the function $N$ forms a Euclidean domain.

Problem 5.

(15 points) Let $\mathbb F_7$ be the finite field of 7 elements.

1. Prove that the polynomial $f(x)=x^7-x+1$ is irreducible in $\mathbb F_7[x]$.

2. Prove that the polynomial $g(x)=x^7+7x^5+6x+8$ is irreducible in $\mathbb Z[x]$.

Problem 6.

(10 points) Let $\alpha=\cos\frac{4\pi}{11}$. Let $E=\mathbb Q(\alpha)$. Compute the Galois group $\operatorname{Gal}(E/\mathbb Q)$.