ALGEBRA PRELIM EXAM January 2006

Let \(G\) be a finite non-abelian group of order \(p^3\) where \(p\) is a prime number. If \(Z\) is the center of \(G\), show that \(G/Z\) is isomorphic to \(\mathbb {Z}/p\mathbb {Z}\oplus \mathbb {Z}/p\mathbb {Z}\).

Classify all groups of order \(57=3\cdot 19\).

Show that \[ \mathbb {Q}/\mathbb {Z}\cong \bigoplus _p(\mathbb {Q}/\mathbb {Z})_{(p)} \] where \(p\) runs over all primes numbers and \((\mathbb {Q}/\mathbb {Z})_{(p)}\) denotes the subgroup of elements of \(\mathbb {Q}/\mathbb {Z}\) whose order is a power of \(p\).

Prove Gauss’ Lemma over the integers: if \(h(x)\in \mathbb {Z}[x]\) such that \(h(x)=f(x)g(x)\) for nonconstant polynomials \(f(x),g(x)\in \mathbb {Q}[x]\), prove that there exist rational numbers \(a\) and \(b\) so that \(F(x)=af(x)\) and \(G(x)=bg(x)\) are in \(\mathbb {Z}[x]\) and so that \(h(x)=F(x)G(x)\).

Let \(R\) be a commutative ring with 1 and \[ 0\longrightarrow K\longrightarrow P\longrightarrow M\longrightarrow 0,\qquad 0\longrightarrow K'\longrightarrow P'\longrightarrow M\longrightarrow 0 \] be short exact sequences of \(R\)-modules where \(P\) and \(P'\) are projective \(R\)-modules. Show that there is an isomorphism of \(R\)-modules \(K\oplus P'\cong K'\oplus P\).

Let \(G\) be the Galois group of the polynomial \(x^4-5\) over the field \(F\). Determine \(G\) in each of the following cases:

1. \(F\) is the real numbers;
2. \(F\) is the rational numbers;
3. \(F\) is the integers modulo 3.

Let \(p\) be a prime number and \(\mathbb {F}_p\) be the field with \(p\) elements. Let \(f(x)=x^p-x-1\in \mathbb {F}_p[x]\) and let \(K\) be the splitting field of \(f(x)\). Show that:

1. if \(\alpha \) is a root of \(f(x)\), then \(\alpha +1\) is a root of \(f(x)\),
2. \(K=\mathbb {F}_p(\alpha )\),
3. the Frobenius automorphism \(\sigma (x)=x^p\) is a non-trivial element of \(\operatorname {Gal}(K/\mathbb {F}_p)\),
4. \(f(x)\in \mathbb {F}_p[x]\) is irreducible.

Let \(K\) and \(F\) be fields such that \(K\) is a finite extension of \(F\). Assume that there are only finitely many subfields \(L\) such that \(F\subset L\subset K\). Show that \(K\) is a primitive extension of \(F\) (i.e. \(K=F(\theta )\) for some \(\theta \in K\)).

Let \(V\) be a finite dimensional vector space over a field \(F\) and \(T:V\longrightarrow V\) be a linear transformation. Show that there is an integer \(n>0\) such that \[ V\cong \operatorname {Ker}(T^n)\oplus \operatorname {Im}(T^n). \]

Let \(A\) be a square complex matrix such that \(A^3=A\). Prove that \(A\) is diagonalizable. Is this true over any algebraically closed field?