Exercise 4, § 14

Let $\Omega$ be an algebraically closed extension of a field K, of infinite transcendence degree over K. a) Show that there exist an infinity of K-endomorphisms of $\Omega$ having as images subfields of $\Omega$ distinct from $\Omega$, and with respect to which $\Omega$ may have a transcendence degree equal to any cardinal up to at most tr . $\deg_K \Omega$. \* In particular, there exist an infinity of distinct $\mathbf{Q}$-isomorphisms of the field C of complex numbers onto subfields of C distinct from C. \* b) Show that for every subextension E of $\Omega$ such that tr . $\deg_K E <$ tr . $\deg_K \Omega$, every K-isomorphism of E onto a subfield of $\Omega$ may be extended to a K-automorphism of $\Omega$. Give examples where tr , $\deg_K E =$ tr . $\deg_K \Omega$ and where there exists a K-isomorphism of E onto a subfield of $\Omega$ which cannot be extended to a K-isomorphism of any extension F of E, contained in $\Omega$ and distinct from E, onto a subfield of $\Omega$.