Chapter 18

Suppose that \(L: K\) is an extension, and that \(\left\\{\alpha_{1}, \ldots, \alpha_{s}\right\\}\) is algebraically independent over \(K\). Show that if \(\beta \in K\left(\alpha_{1}, \ldots, \alpha_{s}\right)\) and \(\beta \notin K\) then \(\beta\) is transcendental over \(K\) (cf. Exercise 5.3).

Suppose that \(K(\alpha): K\) is a simple extension and that \(\alpha\) is transcendental over \(K\). Show that if \(\tau\) is an automorphism of \(K(\alpha)\) which fixes \(K\) then there exist \(a, b, c\) and \(d\) in \(K\) with \(a d \neq b c\) such that $$ \tau(\alpha)=(a \alpha+b) /(c \alpha+d) $$ Conversely show that any such \(a, b, c\) and \(d\) determine an automorphism of \(K(\alpha)\) which fixes \(K\).

Suppose that \(K(\alpha): K\) is a simple extension and that \(\alpha\) is transcendental over \(K\). Let \(\sigma\) be the automorphism of \(K(\alpha)\) which fixes \(K\) and sends \(\alpha\) to \(1 /(1-\alpha)\). Verify that \(\sigma^{3}\) is the identity, and determine the fixed field of \(\sigma\).

Suppose that \(K(\alpha): K\) is a simple extension, that \(\alpha\) is transcendental over \(K\), and that char \(K\) is an odd prime \(p\). Suppose that \(1

Suppose that \(L: K\) is an extension, and that \(L\) is finitely generated over \(K\). Show that the field \(K_{a}\) of elements of \(L\) which are algebraic over \(K\) is finitely generated over \(K\).

Suppose that \(K(x, y): K\) is an extension with \(x\) transcendental over \(K\) and \(x^{2}+y^{2}=1\). Show that \(K(x, y)=K(u)\), where \(u=(1+y) / x\).

Suppose that \(n \geqslant 3\), that \(K(x, y): K\) is an extension with \(x\) transcendental over \(K\) and \(x^{n}+y^{n}=1\) and that \(\operatorname{char} K\) does not divide \(n\). Suppose if possible that \(K(x, y)=s\) (i) Show that there are relatively prime polynomials \(f, g\) and \(h\) in \(K[x]\) such that \(\max (\) degree \(f\), degree \(g\), degree \(h) \geqslant 1 \quad\) and \(f^{n}+g^{n}=h^{n}\) (ii) Show that $$ f^{n-1} \mid(h \mathrm{D} g-g \mathrm{D} h) \quad \text { and } g^{n-1} \mid(h \mathrm{D} f-f \mathrm{D} h) $$ and show (by considering degrees) that this is not possible. R. Hartshorne, Algebraic Geometry, Springer-Verlag, \(1977 .\)