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Determine the number of 5-Sylow subgroups of \(\operatorname {SL}_2(\mathbb {F}_5)\).

Let \(G\) be the subgroup of \(\operatorname {GL}_2(\mathbb {R})\) consisting of matrices of the form \[ \begin {pmatrix} a & c\\ 0 & b \end {pmatrix}. \] Is \(G\) solvable?

Consider the automorphisms \(\sigma ,\tau \) of \(\mathbb {Q}(x)\) with \(\sigma :x\mapsto 1/x\) and \(\tau :x\mapsto 1-x\). What is the order of the group generated by these two elements? Determine the group.

Set \(R=\mathbb {Q}[x]\), and consider the submodule \(M\) of \(R^2\) generated by the elements \((1-2x,-x^2)\) and \((1-x,x-x^2)\). Express \(R^2/M\) as a direct sum of cyclic modules.

Recall that a Hermitian matrix is a complex matrix which equals its conjugate transpose. Determine the conjugacy classes of \(5\times 5\) Hermitian matrices \(A\) satisfying \(A^5+2A^3+3A=6I\).

Determine the number of conjugacy classes of \(4\times 4\) complex matrices satisfying \(A^3-2A^2+A=0\).

Let \(\alpha \) be the positive real root of \(x^6-7\). What is the number of elements of the ring \(\mathbb {Z}[\alpha ]/(\alpha ^2)\)? Is every ideal in this ring principal?

Find the degree of the splitting field of \(x^6-3\) over (i) \(\mathbb {Q}(\sqrt {-3})\) and (ii) \(\mathbb {F}_5\).

Prove that \(x^4+1\) is reducible over any field of positive characteristic.

For \(p\) a prime, determine the Galois group of \(x^p-2\) over \(\mathbb {Q}\). What is its order? Is it abelian?