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Determine the groups of order \(1225=5^2\cdot 7^2\) up to isomorphism.

Let \(p\) be a prime integer. Let \(G\) be a finite \(p\)-group, with identity \(e\), and center \(Z\). If \(N\ne \{e\}\) is a normal subgroup of \(G\), prove that \(N\cap Z\ne \{e\}\).

Let \(K\) be a field and let \(f(x)\in K[x]\) be a monic polynomial of degree \(n\) with \(f(0)\ne 0\). Suppose \(f(x)\) has \(n\) distinct roots in its splitting field, and that the set of roots is closed under multiplication. Determine \(f(x)\).

Let \(M\) be an element of \(GL_n(\mathbb {C})\) that has finite order. Is \(M\) necessarily diagonalizable? Prove or disprove.

Let \(n\) be an integer. Let \(M\) be a square matrix with integer entries, such that the sum of the entries of each row equals \(n\). Prove that \(n\) divides the determinant of \(M\).

A square matrix \(M\) is idempotent if \(M^2=M\). Prove that two idempotent \(n\times n\) matrices over a field are similar if and only if they have the same rank.

Is every ideal of the ring \(\mathbb {Z}\times \mathbb {Z}\) a principal ideal? Prove or disprove.

1. Determine the rank and signature of the real quadratic form \[ x_1x_2+x_2x_3+x_3x_4+x_4x_1. \] (There are different definitions of signature, so state the one you use!)
2. Up to isometry, what is the number of distinct quadratic forms on \(\mathbb {R}^4\)?

Take a regular \(n\)-sided polygon inscribed in a circle of radius 1. Label the vertices \(P_1,\ldots ,P_n\), and let \(\lambda _k\) be the length of the line joining \(P_k\) and \(P_n\) for \(1\leqslant k\leqslant n-1\). Prove that \[ \lambda _1\cdots \lambda _{n-1}=n. \]

Let \(\omega =e^{2\pi i/3}\). Let \(\sigma \) and \(\tau \) be automorphisms of \(\mathbb {C}(x)\) which fix \(\mathbb {C}\) and satisfy \[ \sigma (x)=\omega x,\qquad \tau (x)=1/x. \] Prove that the group \(\langle \sigma ,\tau \rangle \) is isomorphic to the symmetric group \(S_3\), and determine the subfield of \(\mathbb {C}(x)\) that is fixed by this group.