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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.

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.

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.