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18.786 PROBLEM SET 2 Due February 18th, 2016 (1) Recall that for πΉ a number field (i.e., a finite extension of Q), we have AπΉ the topological ring of adeΜles, and its units Aˆ πΉ form the topological group of ideΜles. ˆ p ˆ Qˆ ˆ RΔ 0 Ñ Aˆ is an isomorphism. (a) Show that the canonical map Z Q (b) For πΉ a number field, show that: ΕΊ Λ ˆ ˆ ` Oˆ πΉπ£ zAπΉ {πΉ π£ a place of πΉ is canonically isomorphic to the class group of πΉ , where if π£ is an infinite place, OπΉπ£ :“ πΉπ£ . Here “canonically” means that you should show that such an isomorphism is uniquely characterized by the property that that for each prime ideal p, the composite map: ˆ Z “ Oˆ πΉp zπΉp ãÑ `ΕΊ Λ ˆ `ΕΊ ˆ Λ ˆ ˆ Oˆ OπΉπ£ zAπΉ {πΉ » ClpπΉ q πΉπ£ zAπΉ π£ π£ maps 1 to the ideal class of p. (c) Similarly, show that the profinite completion of: ΕΊ ` Λ ˆ ˆ Oˆ πΉπ£ zAπΉ {πΉ π£ a finite place of πΉ is isomorphic to the narrow 1 class group of πΉ . ` Λ p ˆ R b πΉ Ñ AπΉ is (d) For every number field πΉ , show that the canonical map Z Z an isomorphism. ˆ 2 (2) Recall that Qˆ 2 {pQ2 q has order?8, so Q2 has 7 (isomorphism classes of) quadratic extensions, corresponding to Q2 r πs for π running over a class of coset representatives for the non-squares in Qˆ 2. (a) Using the fact that an element of Zˆ 2 is a square if and only if it is congruent to 1 modulo 8Z2 , show that these coset representatives can be taken to be π “ 2, 3, 5, 6, 7, 10, 14. (b) By the general structure theory of nonarchimedean local fields, Q2 admits a single unramified quadratic extension. Which value of π above does it correspond to? How does this relate to the explicit formula you found last week for the Hilbert symbol for Q2 ? Date: February 16, 2016. 1This is the group of fractional ideals of πΉ modulo principal ideals defined by totally positive elements of ˆ πΉ , i.e., elements π₯ of πΉ ˆ such that for every embedding πΉ ãÑ R, πpπ₯q Δ 0. 1 ? (c) For each π as above, find a uniformizer in the field Q2 r πs, and compute its norm in Q2 . (3) Recall the definition of the quaternion algebra π»π,π associated to π, π P πΎ: it is the πΎ-algebra with generators π and π with relations π2 “ π, π 2 “ π and ππ “ ´ππ. (a) Let πΎ “ Q2 . Show that every π P Q2 admits a square root in π»´1,´1 , i.e., for every π there exists π₯ P π» with π₯2 “ π. (b) Let πΎ be a nonarchimedean local field of odd residue characteristic, and let π, π P πΎ ˆ with Hilbert symbol pπ, πq “ ´1. Show that every element of πΎ admits a square root in π»π,π . (4) Show that a local field πΎ ‰ C contains only finitely many roots of unity. 2