Assignment:
Suppose F, E are fields and F is a subring of E. Prove that the set of ring isomorphisms Q:E-->E is a group Aut(E) under composition *, and that the set Aut(E/F) of isomorphisms Q in G, with Q(f)=f for all f in F is a subgroup of Aut(E).
If E is a field and H is a subgroup of Aut(E), show that the set E^H of elements of E that are fixed (sent to themselves) by every Q in H, is a subring of E that is closed under multiplicative inverses - so is a subfield of E
Show H is a subgroup of Aut(E/E^H) and F is a subgroup of E^Aut(E/F)
Provide complete and step by step solution for the question and show calculations and use formulas.