Chapter 10

Suppose that char \(K=p \neq 0\). Show that every polynomial in \(K[x]\) is separable \((K\) is perfect ) if and only if the Frobenius monomorphism is an automorphism of \(K\).

Suppose that \(f\) is a polynomial in \(K[x]\) of degree \(n\) and that either char \(K=0\) or char \(K>n .\) Suppose that \(\alpha \in K .\) Establish Taylor's formula: $$ f=f(\alpha)+\mathrm{D} f(\alpha)(x-\alpha)+\frac{\mathrm{D}^{2} f(\alpha)}{2 !}(x-\alpha)^{2}+\cdots+\frac{\mathrm{D}^{n} f(\alpha)}{n !}(x-\alpha)^{n} $$

Suppose that \(p\) is a prime number of the form \(4 n+1\). Show that there exists \(k\) such that \(k^{2}+1=0(\bmod p)\). Show that \(p\) is not a prime in \(\mathbb{Z}+\mathrm{i} \mathbb{Z}\) and show that there exist \(u\) and \(v\) in \(\mathbb{Z}\) such that \(u^{2}+v^{2}=p\)

Suppose that char \(K=p>0\) and that \(L: K\) is a totally inseparable algebraic extension: that is, every element of \(L \backslash K\) is inseparable. Show that if \(\beta \in L\) then its minimal polynomial over \(K\) is of the form \(x^{p^{n}}-\alpha\), where \(\alpha \in K\).