Chapter 9

Show that every algebraic extension has a normal closure.

Suppose that \(L: K\) is algebraic. Show that there is a greatest intermediate field \(M\) for which \(M: K\) is normal.

Suppose that \(L: K\) and that \(M_{1}\) and \(M_{2}\) are intermediate fields. Show that if \(M_{1}: K\) and \(M_{2}: K\) are normal then so are \(K\left(M_{1}, M_{2}\right): K\) and \(M_{1} \cap M_{2}: K\)

Suppose that \(N: L\) and \(N^{\prime}: L\) are two normal closures of \(L: K\). Show that there is an isomorphism \(j\) of \(N\) onto \(N^{\prime}\) such that \(j(l)=l\) for \(l \in L\).

Suppose that \(L: K\) is a finite normal extension and that \(f\) is an irreducible polynomial in \(K[x]\). Suppose that \(g\) and \(h\) are irreducible monic factors of \(f\) in \(L[x]\). Show that there is an automorphism \(\sigma\) of \(L\) which fixes \(K\) such that \(\sigma(g)=h\).

Suppose that \(L: K\) is algebraic. Show that the following are equivalent: (i) \(L: K\) is normal; (ii) if \(j\) is any monomorphism from \(L\) to \(\bar{L}\) which fixes \(K\) then \(j(L) \subseteq L\) (iii) if \(j\) is any monomorphism from \(L\) to \(\bar{L}\) which fixes \(K\) then \(j(L)=L\)