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(a) If $|𝐺|=12$ and $𝐻\le 𝐺$ has order $6$, prove that ${𝑔}^2\in 𝐻$ for every $𝑔\in 𝐺$.

(b) Prove that ${𝐴}_4$ has no subgroup of order $6$.

If $𝐻$ has finite index $𝑛$ in a group $𝐺$, prove that there is a normal subgroup $𝐾\text{◃}𝐺$ such that

Is $\mathbb{Z}[1/2]$ finitely generated as a $\mathbb{Z}$-module?

Let $𝑅$ be Noetherian, let $𝐼\subseteq 𝑝$ be ideals, and suppose $𝑝$ is the only prime ideal containing $𝐼$. Prove that

for some $𝑚>0$.

Let $𝐹$ be a field and let $𝐴\in {𝑀}_{𝑛}(𝐹)$ satisfy ${𝐴}^2={𝐼}_{𝑛}$ and $𝐴\ne {𝐼}_{𝑛}$. Prove that $-1$ is an eigenvalue of $𝐴$.

For which positive integers $𝑒$ is ${𝑥}^2+𝑥+1$ irreducible over ${𝔽}_{2^{𝑒}}$?

Let $𝐾/𝐹$ be finite Galois. Suppose $𝛼\in 𝐾$, $0\ne 𝛽\in 𝐹$, and

for some $𝜎\in Gal(𝐾/𝐹)$. Prove that $𝐾$ has positive characteristic.

Prove that $\mathbb{Q}\left(\sqrt{3+2\sqrt{2}}\right)$ has degree four over $\mathbb{Q}$ and is Galois over $\mathbb{Q}$, and compute its Galois group.

Let $𝐺$ be a finite group and let $𝐻$ be a maximal subgroup of $𝐺$. Prove that the number of elements of $𝐺$ contained in every conjugate of $𝐻$ is at most

Let $𝛼\ne -1$ be a root of ${𝑥}^5+1\in \mathbb{Q}[𝑥]$. Is $\mathbb{Q}(𝛼)/\mathbb{Q}$ Galois?