1. Let E/F be an algebraic extension and σ:E→Ean embedding of E
into itself that is the identity on F. Show that σmust be an isomorphism
of Eonto itself.
2. If E/F is an extension, we denote by Aut(E/F ) the group of automor-
phisms of Eover Fi.e. the set of isomorphisms from Eto itself that are the
identity on F, with composition as the group law. Write down Aut(E/F )
explicitly for the following extensions: (Justify your answers.)
(a) Q(ω, 21/3)/Q, where ωis a primitive cube root of unity.
(b) Q(ζn)/Q, where ζnis a primitive nth root of unity.
3. Find the splitting fields of the following polynomials over Q:
(a) X5−11.
(b) X6+X3+ 1.
4. Let αbe a root of X3+X2+X+ 2. Write 1
α−1as a quadratic polynomial
in αwith coefficients in Q.
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