Mathematics 400c
Homework (due April 28)
A. Hulpke
71) This problem is to give an alternative proof of the infinitude of primes due to E U LE R:
Assume there would be only finitely many primes. Show that the product formula for the
value ζ(1) equals a divergent sum with a finite (and thus convergent) product.
72) Let f (x) ∈ Q[x] be a polynomial with rational coefficients and α ∈ C such that f (α) =
0. Show that there exists an integer d, such that d · α is an algebraic integer.
Hint: In the expression f (x) replace x by x/d to eliminate denominators. For this new
polynomial g(x/d) then consider the expression d deg(g) · g(x/d).
73∗ ) Let α, β be algebraic integers (i.e. there are monic polynomials f (x), g(x) ∈ Z[x] with
f (α) = 0 = g(β)). Show that α + β is an algebraic integer.
Hint: Construct a monic polynomial p(x) ∈ Z[x] with integer coefficents such that p(α +
β) = 0.
74) Determine whether α|β in the Gaussian integers Z[i] and – if so – determine the
quotient β/α as an element of Z[i]:
a) α = 3 + 5i, β = 11 − 8i.
b) α = 3 + 5i, β = 21 + i.
c) α = 3 − 39i, β = 42 − 36i.
75∗ ) Find (for example by searching on the web) an example of a theorem that has been
proven only under the assumption that the Riemann hypothesis holds (and thus really is
not yet proven).
Problems marked with a ∗ are bonus problems for extra credit.