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Math 627 Homework #5 November 30, 2014 Due Tuesday, December 9 1. Let F be a finite field with q elements, let A = F [T ] be a polynomial ring in one variable, and let A+ be the elements of A which are monic. We can define the Dedekind zeta function for A as, X 1 , s ∈ C, Re(s) > 1. ζA (s) = s N (a) a⊆A a6=(0) (a) Show that ζA (s) = X a∈A+ 1 q (deg a)s . 1 . 1 − q 1−s (c) For d ≥ 1, let ad denote the number of monic irreducible polynomials of degree d. Show that ∞ −ad Y −ds 1−q . ζA (s) = (b) If we group terms by degree, use the sum in (a) to show that ζA (s) = d=1 −s (d) If we let u = q , then combining (b) and (c), we have ∞ Y −a 1 = 1 − ud d . 1 − qu d=1 Taking a logarithmic derivative with respect to u, prove that ∞ X dad ud qu = . 1 − qu 1 − ud d=1 (e) Use (d) to prove that X dad = q n . d|n Then use MoĢbius inversion to prove an = 1X µ(d)q n/d . n d|n √ 2. Use√ the Class Number Formula to find the class numbers of the fields Q( −7), √ Q( −10), and Q( −23). √ 3. Use√the Class Number Formula to find the √ class numbers √ of the fields Q( 7) and Q( 15). You may use the facts that 8 + 3 7 and 4 + 15 are fundamental units in these fields. 1