Mathematics 400c Homework (due April 28) 71) A. Hulpke

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.