# HW 11, due Wed December 7

• Use terms of the continued fraction expansion of $\sqrt{7}$ to find a solution to Pell’s equation $m^2 - 7 n^2 = 1$.
• Calculate the Bernoulli number $B_{6}$.
• Check that the Bernoulli number formula for $1^3 + 2^3 + \cdots+ (n-1)^3$ gives the usual formula $(n (n-1)/2)^2$.
• Use the functional equation for the Riemann zeta function to calculate $\zeta(-3)$.
• If P1 and P2 are disjoint sets of primes and if both have a well-defined Dirichlet density, then show that densities satisfy d(P1+P2) = d(P1)+d(P2).
• If P is a finite set of primes, show that it has Dirichlet density 0.

# Exam Schedule, Dec 14

The final exam will be held

Wednesday December 14, 2016, 4:00PM – 5:50PM, in
103 Alexander J. Allen Hall (our usual room).

There will be extra office hours

• Monday December 12, 2-3pm
• Wednesday December 14, 2-3pm.

# HW 10, due Mon Nov 21

• MNT Chapter 1 (page 16) problem 38
• MNT Chapter 13, problem 10.  You may quote results from the book that have not been covered in class. For example, compute the discriminant $\Delta(\alpha,\alpha')$, where $\alpha = a + b \sqrt{d}$ where the integer $\alpha$ is given in Prop. 13.1.1 and compare with the formula of Proposition 13.1.2.

# HW 9, due Wed Nov 16

• MNT 12.1
• MNT 12.2  You may compute the discriminant with respect to the basis $1, \sqrt{2}, (1+\sqrt{5})/2, (1+\sqrt{5})\sqrt{2}/2$.
• MNT 12.4.  In this problem “integer” means algebraic integer, the conjugate of $\alpha = a + b \sqrt{d}$ means  $\alpha' = a- b\sqrt{d}$.  Also, $|\alpha|$ is the absolute value of $\alpha$ (viewed as a complex number).  Hint: can you bound the number of quadratic monic polynomials with integer coefficients?
• MNT 12.6
• MNT 12.9.  You may work this for an irreducible cubic rather than a reducible cubic, if you prefer, using 12.1.4.

# No office hours Wed Nov 9

There will be no office hours on Wednesday or after class on Monday and Wednesday.  Alex Yarosh will give the lecture that day.  Hiruni Pallage’s office hours in the MAC will continue as usual.

# HW 8, due Wednesday Nov 9 (updated)

• Let $F=F_2(t)$ be a field with four elements, where $t^2+t+1=0$.  Compute the norm of every element of F.
• Find the values of the parameter $\lambda$ such that $x^2 - y^2 + \lambda z^2=0$ has a singular point in the projective plane.
• Suppose that the zeta function has the form $(1+u+ 3 u^2)/(1-4 u + 3 u^2)$.  (a) What is $q$, the cardinality of the finite field?  (b) Find how many points there are in $F_{q^3}$.
• A zeta function has the form $(1 + a u + 7 u^2)/(1 - 8 u + 7 u^2)$, for some integer a.  (a) What is $q$, the cardinality of the finite field?  (b) If $N_1 = 9$, then what is the integer a?

# HW 7 due Nov 2 2016

• Count the number of solutions to the equation $x^2 + 3 y^2 \equiv 1 \mod p$ in two-dimensional projective space working in the field with p elements, assuming p>3.  Use the procedure described in class to convert to a homogeneous polynomial to find the solutions at infinity.
• problem 10.2 of MNT
• problem 10.3 of MNT
• problem 10.7 of MNT
• compute the trace of every element of the field $F_2(t)$, where $t^2 + t + 1=0$.