Help with GAlois

If we write Q~C for the algebraic closure of Q inside C.

(a) We know that C is algebraically closed. Show that Q~ is also algebraically closed.

(b) Show that Q[x] contains irreducible polynomials of any degree n.

(c) Show that for any n, QQ~ has a subfield of degree n.

(d) Show that Q~ is not a finite extension of Q ( proving that field does not have finite degree

over Q requires a bit more number theory. There are other subfields you can build of

arbitrary degree, where it's easier to show it's of arbitrary degree.