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.