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Determine, up to isomorphism, the groups of order 75.

Let \(G\) be a nonabelian simple group. Prove that \(\operatorname {Aut}G\) contains a normal subgroup isomorphic to \(G\).

Is the group of positive rational numbers under multiplication torsion-free? Is it a free abelian group?

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

Let \(A\) be a square matrix with coefficients in \(\mathbb {C}\). Prove that there exist polynomials \(s(t)\) and \(n(t)\) with coefficients in \(\mathbb {C}\), such that \(s(A)\) is diagonalizable, \(n(A)\) is nilpotent, and \(A=s(A)+n(A)\).

Find all the maximal ideals of \(\mathbb {Z}[\sqrt {-13}]\) that contain 7, and prove that their product is precisely the ideal \((7)\).

Let \(p\) be a prime number, \(n\) a positive integer, and \(V\) the field \(\mathbb {F}_{p^n}\) viewed as a vector space over \(\mathbb {F}_p\). Prove that the Frobenius map \(\varphi :V\longrightarrow V\), assigning \(v\) to \(v^p\), is diagonalizable over \(\mathbb {F}_p\) if and only if \(n\) divides \(p-1\).

Determine the minimal polynomial of \(\sin (2\pi /5)\) over \(\mathbb {Q}\).

Determine the Galois group of \(x^4-16x^2+4\) over \(\mathbb {Q}\).

Let \(L/K\) be a Galois extension of fields of characteristic zero. Suppose \(f(x)\) is an irreducible polynomial of degree 5 in \(K[x]\). If \(f(x)\) has no roots in \(L\), prove that it is irreducible in \(L[x]\).