Chapter 12

Suppose that \(L: K\) is an extension and that \(L\) has \(p^{n}\) elements. Show that \(|K|=p^{d}\), where \(d \mid n .\) Conversely, if \(d \mid n\), show that \(\left(x^{p^{d}}-x\right) \mid\left(x^{p^{n}}-x\right)\) and deduce that \(L\) has exactly one subfield with \(p^{d}\) elements.

Suppose that \(a\) and \(b\) are positive integers with highest common factor \(d\). Show that $$ \mathbb{Z}_{a} \times \mathbb{Z}_{b} \cong \mathbb{Z}_{d} \times \mathbb{Z}_{a b / d}. $$

Suppose that \(G\) is an abelian group. Show that the set \(T\) of elements of finite order is a subgroup of \(G\) and that every element of \(G / T\), except the identity, is of infinite order.

Suppose that \(G\) is a finitely generated abelian group every element of which, except the identity, has infinite order. Show that \(G \cong \mathbb{Z}^{s}\), where \(s\) is defined by the property that \(G\) is generated by \(s\) elements, but is not generated by \(s-1\) elements.

Suppose that \(G\) is a finitely generated abelian group. Show that \(G \cong \mathbb{Z}^{s} \times T\), where \(T\) is a finite group.

Suppose that \(p\) and \(q\) are primes and that \(p