# 189-570A: Higher Algebra I

## Due: Friday, October 15.

1. From Chapter V, (Algebraic Extensions). Problem 22.

2. Extra Credit: Chapter V, Problem 23. This exercise is challenging and quite rewarding, so I recommend it highly. Those of you who were in the number theory course last year will find it helpful to recall the manipulations we followed to relate the Riemann zeta-function to questions about distributions of primes. Note that the squiggle sign in (b) means that the ratio of the two sides tends to 1 as m tends to infinity. As an instructive complement to the question, assuming Hasse's theorem about the zeroes of Z(t) for an elliptic curve, what can you deduce about the asymptotics of the function piq(m)? Explain why Hasse's theorem is viewed as the analogue of the Riemann hypothesis for the function field of the elliptic curve.

3. From Chapter VI (Galois Theory), problem 1. (a) (e) (f) (m) (n).

4. From Chapter VI (Galois Theory), problem 4.

5. From Chapter VI (Galois Theory), problem 10.

6. From Chapter VI (Galois Theory), problem 11.

7. From Chapter VI (Galois Theory), problem 14.