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Prove that every group of order 15 is cyclic.

Determine, up to isomorphism, the groups of order 18.

Let \(\mathbb {F}\) be a finite field with \(q\) elements. Determine the number of \(2\times 2\) nilpotent matrices with entries from \(\mathbb {F}\).

Let \(M\) and \(N\) be finitely generated modules over a commutative local ring \(A\). If \(M\otimes _A N=0\), prove that \(M=0\) or \(N=0\). Is this true if \(A\) is not local? Prove or disprove.

Let \(A\) be a commutative ring and let \[ 0\longrightarrow L\longrightarrow M\longrightarrow N\longrightarrow 0 \] be a short exact sequence of \(A\)-modules. If \(L\) and \(M\) are injective, show that the sequence splits, and that \(N\) is also injective.

Let \(\mathbb {F}\) be a field and let \(A\) be the polynomial ring \(\mathbb {F}[x_1,x_2,\ldots ,x_n]\). Regard \(\mathbb {F}\) as an \(A\)-module via the map \(A\longrightarrow A/(x_1,x_2,\ldots ,x_n)A\). Prove that \[ \operatorname {Tor}^{A}_{i}(\mathbb {F},\mathbb {F})\cong \operatorname {Ext}^{i}_{A}(\mathbb {F},\mathbb {F})\cong \bigwedge ^{i}(\mathbb {F}^{n}) \quad \text {for each } i\geq 0. \]

Determine, in each case, whether the given ring is a unique factorization domain:

1. \(\mathbb {Z}[\sqrt {-1}]\),
2. \(\mathbb {Z}[\sqrt {-5}]\),
3. \(\mathbb {Z}[2\sqrt {2}]\)

Compute the Galois group of the polynomial \((x^2-2)(x^3-2)(x^3-3)\) over the field \(\mathbb {Q}(\sqrt {-3})\).

Given an integer \(n\), prove that there exists a monic polynomial with integer coefficients whose Galois group over \(\mathbb {Q}\) is \(S_n\).

Let \(\alpha \) be a complex root of \(x^6+x^3+1\). Determine all field homomorphisms \(\mathbb {Q}(\alpha )\longrightarrow \mathbb {C}\).