HW due Friday April 20

1. Show that Q[cube root of two, omega] is a simple extension, where omega = exp(2 pi i/3).

2. Let F = K[u], a finite simple extension, where u satisfies an irreducible polynomial p(x) in K[x].

a. Let u' be another root in F. Show there exists an automorphism of F taking u to u'.

b. Show |Gal(K/F)| is at most deg(f) deg(p).

c. Determine Gal(Q[cube root of two, omega] / Q), meaning, for each group element write down what it does to those two field elements.

3. Let F > K denote a field extension, with R a ring in between the two.

a. Give an example where R is not a field.

b. If F is algebraic over K, show R is indeed a field.

4. Let R be a domain, K the fraction field of R (so, not matching the notation in #3), F an extension of K, and f in F algebraic over K.
Show there exists a nonzero r in R such that rf is integral over R.

5. What are all the rings between Z and Q?
Which ones are finitely generated as Z-algebras?
Which ones are finitely generated as Z-modules?