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Prove that the additive group of \(\mathbb {Q}\) is not a projective \(\mathbb {Z}\)-module.

Determine the splitting field of the polynomial \(x^p-x-a\) over \(\mathbb {F}_p\) where \(0\ne a\in \mathbb {F}_p\). Show that the Galois group is cyclic.

What is the Galois closure of \(\mathbb {Q}(\sqrt {1+\sqrt {2}})/\mathbb {Q}\)?

Prove that \(\mathbb {Q}(\sqrt [3]{2})\) is not a subfield of any cyclotomic field over \(\mathbb {Q}\).

Find the Galois group of \(f(x)=x^4+2x^2+x+3\).

Let \(R=\mathbb {F}_3[x]\). Consider the \(3\times 3\) matrix \[ \begin {bmatrix} 0 & x & x\\ 1 & x & 1+x\\ 1+x^2 & x^2 & 1+x^2 \end {bmatrix} \] Let \(\varphi :R^3\longrightarrow R^3\) be the \(R\)-module homomorphism given by this matrix. Write the cokernel of \(\varphi \) as a direct sum of cyclic modules.

Show that there is no simple group of order \(552=2^3\cdot 3\cdot 23\).

Suppose that \(k\) is the field with 3 elements and let \(R=k[x]\). Identify nine different (up to isomorphism) \(R\)-modules \(M\) such that \(|M|=9\).

Suppose \(\varphi :G\longrightarrow H\) is a group homomorphism and \(H\) is a group of order 33. Further suppose that \(N\supseteq \ker \varphi \) is a subgroup of \(G\). Prove that \(N\) is normal.

Suppose \(k\) is a field and suppose that \(f(x)\in k[x]\) has degree \(n\). Prove that \(f(x)\) has no repeated factors in \(k[x]\) if and only if all \(n\times n\) matrices over \(k\) with characteristic polynomial \(f(x)\) are similar.