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18.786 PROBLEM SET 3 Due February 25th, 2016 (1) Construct (with proofs) an abelian extension πΈ{πΉ of number fields such that πΈ does not embed into any cyclotomic extension of πΉ , i.e., there does not exist an integer π such that πΈ embeds into πΉ pππ q. (2) Let πΎ β° C be a local field of characteristic β° 2. For π, π P πΎ Λ , π»π,π denotes the corresponding Hamiltonian algebra over πΎ. You can assume all good properties of Hilbert symbols in this problem (since we have not proved them yet for residue characteristic 2 and πΎ β° Q2 ). ? (a) Show that π»π,π » π»π,π if and only if π β π P πΎ Λ {π pπΎr πsΛ q. (b) Show that the isomorphism class of π»π,π depends only on the Hilbert symbol pπ, πq. (c) Give another proof of (a slight extension of) that exercise from last week: every π₯ P πΎ admits a square root in π»π,π for all pairs π, π P πΎ Λ . (d) Show that any noncommutative 4-dimensional division algebra π» over πΎ is a Hamiltonian algebra. Deduce that there is a unique 4-dimensional division algebra over πΎ. (3) In this problem, we will examine how far the tame symbol (defined in the first problem set) can take us in local class field theory. (a) Let π Δ 1 be an integer and let πΎ be a field of characteristic prime to π. Let ππ Δ πΎ Λ denote the subgroup of πth roots of unity. Suppose that |ππ | β π, i.e., πΎ admits a primitive πth root of unity.1. Construct a canonical isomorphism: HompGalpπΎq, ππ q » πΎ Λ {pπΎ Λ qπ where Hom indicates the abelian group of continuous morphisms.2 (b) Now suppose that πΎ is a nonarchimedean local field. Let π denote the order of the residue field π β OπΎ {p of πΎ. Suppose that π divides π ´ 1 (e.g., π β 2 and π is odd) for the remainder of this problem. Show that every element of ππ lies in the ring of integers of πΎ. Show that the mod p reduction map: Updated: February 25, 2016. 1I.e., suppose that there exists an isomorphism π » Z{πZ, but we do not fix such an isomorphism at π the onset. In what follows, canonical means that you should not choose an isomorphism ππ » Z{πZ in making your constructions (though you are welcome to use it in the course of proving claims about your constructions) 2A small hint: first identify the left hand side with the set of Galois extensions πΏ{πΎ equipped with an embedding GalpπΏ{πΎq ãÑ ππ (up to isomorphism). 1 ππ Ñ tπ₯ P π Λ | π₯π β 1u is an isomorphism. Deduce that |ππ | β π. (c) Construct a canonical isomorphism between π Λ {pπ Λ qπ and ππ . (d) Show that the composition: tame symbol πΎ Λ Λ πΎ Λ ÝÝÝÝÝÝÝÑ π Λ Ñ π Λ {pπ Λ qπ » ππ induces a bimultiplicative pairing: πΎ Λ {pπΎ Λ qπ Λ πΎ Λ {pπΎ Λ qπ Ñ ππ that is non-degenerate in the sense that the induced map: πΎ Λ {pπΎ Λ qπ Ñ HompπΎ Λ , ππ q is an isomorphism. (4) (a) Let πΏ{πΎ be an unramified extension of local fields of degree π. Show that πΎ Λ {π pπΏΛ q is cyclic of order π. (b) Let πΏ{πΎ be a totally ramified extension of degree π. Assume π divides π ´ 1, with π the order of the residue field of πΎ (which is also the residue field of πΏ). Show that the canonical map: Λ Λ Λ OΛ πΎ {π pOπΏ q Ñ πΎ {π pπΏ q is an isomorphism. Show that the reduction map: Λ Λ Λ π OΛ πΎ {π pOπΏ q Ñ π {pπ q is well-defined and an isomorphism. Deduce that πΎ Λ {π pπΏΛ q canonically isomorphic to ππ . (c) Briefly, what is the relationship between this problem and the previous one? (5) In the next exercise (which is long but locally easy), assume any standard results you like from Galois theory. The point is to get a bit more comfortable with the profinite Galois group. Let πΊ be a finite group and πΎ a field. A πΊ-torsor over3 πΎ is a commutative πΎ-algebra πΏ with an action of πΊ by πΎ-automorphisms, such that the canonical map: ΕΊ πΏbπΏÑ πΏ πΎ πPπΊ π b π ÞÑ ppπ ¨ πq ¨ πqπPπΊ is an isomorphism of πΎ-algebras. Here in the formula on the right, we are giving the coordinates of the result of applying our function, and π ¨ π means we act on π P πΏ by π P πΊ, while the second ¨ is multiplication in the algebra πΏ. (a) Show that any πΊ-torsor πΏ is eΜtale as a πΎ-algebra. 3In algebraic geometry, we would rather say over SpecpπΎq. 2 Ε (b) Show that πΏ β πPπΊ πΎ is a πΊ-torsor over πΎ, where the πΊ-action permutes the coordinates. This πΊ-torsor is called the trivial πΊ-torsor. (c) If πΏ is Galois extension of πΎ (in particular, πΏ is a field) with Galois group πΊ, show that πΏ is a πΊ-torsor over πΎ. (d) We now fix a separable closure πΎ π ππ of πΎ. Let us say a rigidification of a πΊtorsor πΏ as above is the datum of a map π : πΏ Ñ πΎ π ππ of πΎ-algebras. Show that every πΊ-torsor πΏ admits a rigidification. (e) Note that πΊ acts on the set of rigidifications of πΏ through its action on πΏ. Show that this action is simple and transitive. (f) Given a rigidified πΊ-torsor π : πΏ Ñ πΎ π ππ , show that the only automorphism π of πΏ as a πΎ-algebra that commutes with both the πΊ-action and with π (i.e., πpπpπqq β πpπq for all π P πΏ) is the identity. (g) Let GalpπΎq :β AutπΎβalg pπΎ π ππ q be the absolute Galois group of πΎ, considered as a profinite group. Show that the set of continuous homomorphisms π : GalpπΎq Ñ πΊ are in canonical bijection with the set of isomorphism classes of rigidified πΊ-torsors over πΎ. As a hint, here Ε is one direction in the construction: given π, we take πΏ to be the subalgebra of πPπΊ πΎ π ππ consisting of elements of the form pππ qπPπΊ such that for every πΎ P GalpπΎq, πΎ ¨ππ β ππpπΎq¨π (here πΎ ¨ππ indicates the action of GalpπΎq on πΎ π ππ ), and the rigidification to be projection onto the coordinate corresponding to 1 P πΊ. (h) For a continuous homomorphism π : GalpπΎq Ñ πΊ, show that the resulting πΎalgebra πΏ is a field if and only if π is surjective, and in this case, is the Galois subfield of πΎ π ππ with Galois group πΊ corresponding (under infinite Galois theory) to this quotient πΊ of GalpπΎq. (i) Show that Ε the trivial homomorphism π : GalpπΎq Ñ πΊ corresponds to the πΊtorsor πPπΊ πΎ with rigidification induced by the projection onto the coordinate for 1 P πΊ. (j) (Not for credit.) If you know the fundamental group π1 pπ, π₯q of a (sufficiently nice) topological space π, then formulate a notion of πΊ-torsor over π and show that if π is connected, a πΊ-torsor with a lift of the basepoint is the same as a homomorphism π1 pπ, π₯q Ñ πΊ. (k) (Not for credit.) Invent the eΜtale fundamental group for schemes. Formulate all the main results of Grothendieckβs SGA I. Bonus non-credit for proving all those results. 3