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ring isomorphism

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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)

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Solution Summary

This solution provides the steps to prove a given statement about a set of ring isomorphisms.