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Suppose \(G\) is a group acting transitively on a finite set \(S\), \(|S|\geq 2\). Prove that there exists an element \(\sigma \in G\) such that \(\sigma (s)\ne s\), for all \(s\in S\).

Let \(S^1\) be the circle group, i.e., the group of all complex numbers of norm 1 with multiplication. If \(A\) is a finite abelian group, define the dual group \(\widehat {A}\) to be the multiplicative group of all group homomorphisms \(A\to S^1\). Prove that \(A\cong \widehat {A}\).

Let \(R\) be a commutative ring with 1 and \(M_n(R)\) the ring of \(n\times n\) matrices with coefficients in \(R\). Prove that every ideal of \(M_n(R)\) is of the form \(M_n(I)\), for some ideal \(I\) of \(R\).

Let \(M\) be a \(5\times 5\) matrix with real entries. Suppose \(M\) has finite order and \(\det (M-I_5)\ne 0\). Find \(\det (M)\).

Let \(\phi \) denote the Frobenius map \(x\mapsto x^p\) on the finite field \(\mathbb {F}_{p^n}\). Determine the Jordan canonical form (over \(\overline {\mathbb {F}}_p\)) for \(\phi \) regarded as an \(\mathbb {F}_p\)-linear transformation of \(\mathbb {F}_{p^n}\).

Determine the splitting field and the Galois group for the polynomial \(x^3-2\) over \(\mathbb {Q}\).

Show that the polynomial \(x^4+1\) is reducible modulo every prime \(p\).

Let \(K\subseteq L\) be fields, and let \(f(x)\) be an irreducible polynomial in \(K[x]\). If there exists \(a\) in \(L\) with \(f(a)=0=f(a^2)\), prove that \(f(x)\) splits in \(L[x]\).

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

Show that \(\mathbb {Q}\otimes _{\mathbb {Z}}\mathbb {Q}\cong \mathbb {Q}\otimes _{\mathbb {Q}}\mathbb {Q}\) as left \(\mathbb {Q}\)-modules.