Algebraic Number Theory
Math 514A Fall 2007
Problem Set 9
Due: Tuesday, Nov. 13th
1. For any prime p > 2, show that Q(ζp)/Qhas a unique subextension of degree 2,
say Q(√d)/Q, and describe d.
2. Show that the ideals (x+ζi
py)⊂Z[ζp] for i= 0, . . . , p −1 are pairwise relatively
prime.
3. For any unit ∈Q(ζp), show that there exists an 0∈Q(ζp+ζ−1
p) and r∈Zsuch
that =ζr
p0.
4. Let Qn=Q[µn] be the nth cyclotomic number field. Complete the proof of the
assertion that OQn=Z[µn]. First for prime number power n=pα, then in general.
5. Let F/K be a function field of one variable with genus g, and Kbe a canonical
divisor of F. Use the Riemann-Roch Theorem to show:
a) If Dis a positive divisor of degree at least g+ 1, then there is a nonconstant
function in L(D).
a) If g≥2 and m≥2, then dim L(mK ) = (g−1)(2m−1).
6. Let Cbe a curve of genus 2 over an algebraically closed field k. Let Obe a k-point
of C, and let C(2) be the symmetric square of C, i.e. the set of unordered pairs of
points of C, and denote pairs {P1, P2}in C(2) by P1+P2. Using the Riemann-Roch
Theorem, show the following:
(a) If P1, P2, Q1, Q2∈C, then there exist R1, R2∈Csuch that P1+P2+Q1+Q2
is linear equivalent to R1+R2+ 2Oas divisors on C.
(b) Suppose the elment R1+R2∈C(2) in part (a) is unique, show that P1+
P2+Q1+Q2is not linearly equivalent to 2O+K, where Kis the canonical
divisor on C.