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

How many elements of \(S_7\) commute with the permutation \((123)(45)(67)\)?

Determine all prime ideals of the ring \(\mathbb {R}[x,ix]\), where \(x\) is an indeterminate over \(\mathbb {R}\), and \(i^2=-1\).

Let \(R=\mathbb {Z}[i]\) and \(S=\mathbb {Z}[\sqrt {-3}]\). Does there exist a ring homomorphism \(R\longrightarrow S\)? Does there exist a ring homomorphism \(S\longrightarrow R\)?

Determine, up to conjugacy, all real \(5\times 5\) matrices with characteristic polynomial \(x(x^2+1)^2\).

Let \(n\) be a positive integer, \(V\) a \(\mathbb {C}\)-vector space of rank \(n\), and \(T:V\longrightarrow V\) a \(\mathbb {C}\)-linear map with the property that for each \(c\in \mathbb {C}\) the eigen space \(\{v\in V\mid T(v)=cv\}\) has rank at most 1. Prove that there exists a \(w\in V\) such that the set \(\{w,T(w),\ldots ,T^{n-1}(w)\}\) is a basis for \(V\).

Prove that every real matrix that commutes with the matrix \[ A:=\begin {bmatrix} 1 & 2\\ 3 & 4 \end {bmatrix} \] is of the form \(rI+sA\), for some real numbers \(r,s\).

Determine the minimal polynomial of \(\pi \) over the field \(\mathbb {Q}((\pi ^2+1)/(\pi -1))\).

(Use, without proof, that \(\pi \) is transcendental!)

Let \(\alpha =2^{1/5}+5^{1/2}\). Determine the extension degree of \(\alpha \) over \(\mathbb {Q}\).

Determine the minimal polynomial of \(\sqrt {4+\sqrt {7}}\) over \(\mathbb {Q}\). Determine the Galois group of this polynomial.