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Classification of Quaternion Algebras Chapter 5 Classification of quaternion algebras Due to time constraints, this chapter contains less than what I hoped. For instance, some proofs are omitted, and some are just done over Q. Still, I think the arguments over Q are nice, where one go though things a bit more explicitly, and use tricks to simplify things that don’t quite work in the general case. I may add more to this in the future, but as far as expanding these notes go (which I am slowly doing though the course is over), my priorities now are toward later chapters. 5.1 Quaternion algebras over local fields Here we classify quaternion algebras over F where F is a local field of characteristic 0, i.e., F is a p-adic field or R or C. The case where F is archimedean was already done in Section 2.6, so it suffices to work out the classification when F is p-adic. For Fp-adic, let $F denote a uniformizer of F . This will be a special case of Theorem 2.7.3. Here is the general statement. Theorem 5.1.1. Let F be a local field of characteristic 0. If F = C,thentheonlyquaternion 1, 1 F M ( ) B = = − − F = algebra over (up to isomorphism) is 2 C .Put H R if R and u,$F B = if F is p-adic where u ⇥ is a nonsquare. If F = C,then,uptoisomorphism, F 2OF 6 there are exactly two quaternion algebras over F : M (F ) and the unique quaternion division 2 algebra B. Implicit in the statement in the p-adic case is that B does not depend upon the choice of u or the uniformizer $F . Lemma 5.1.2. Let F be a p-adic field, and B/F aquaternionalgebra.ThenB contains the unique unramified quadratic extension K/F . Proof. This is a special case of Corollary 4.3.5, which we did not prove. Either B M (F ) ' 2 or B is division. If B M (F ), then the result is true by Proposition 2.4.1.SoassumeB ' 2 is division. For simplicity, we will just complete the proof when F = Qp with p odd. In this case, there are 3 nontrivial square classes represented by u, p,andup, where u Z⇥ is not a 2 p square (cf. Proposition 1.2.12). Recall Qp(pu) is the unique quadratic unramified extension of Qp. 139 QUAINT Chapter5: Classificationofquaternionalgebras KimballMartin We may thus assume B = a,b where a, b u, p, up . It suffices to show we can take Qp 2{ } pu1,pu2 a = u. If not, then we may assume a, b p, up ,soB = where u1,u2 1,u . Qp 2{ } 2 2{ } a,b ab,b p u1u2,pu2 u1u2,pu2 But since − (cf. Corollary 3.3.3), we have B = − = − . Since F ' F F F B is not split, it must be that u u is in the same square class as u, so indeed we can take − 1 2 a = u. Exercise 5.1.1. Prove the above lemma when F = Q2. This takes care of the lemma when F is a “prime” p-adic field, i.e., F is some Qp.We omit the proof in the case F is an extension of some Qp. Proof of theorem. It suffices to show that there is a unique quaternion division algebra in the p-adic case, and that it is given in the above form. Again, for simplicity, we will just complete the proof when F = Qp with p odd. Then, from the proof of the lemma we know any quaternion division algebra over Qp is of the form B = u,b , where b u, p, up .NoteB = u,u is split by Proposition 3.3.7 since the norm Qp 2{ } Qp map Qp(pu)⇥ Q⇥ is surjective on integral units (Corollary 1.2.15). ! p Thus either B = u,p or B = u,up . Itsuffices to show that these are isomorphic. From Qp Qp a,b a, ab the fact that = − (Corollary 3.3.3), they are isomorphic if 1 is a square in Qp, F F − i.e., if p 1 mod 4. ⌘ Assume that 1 is not a square in Qp, in which case we may take u = 1. Then we − − want to show the associated restricted norm forms x2 py2 pz2 and x2 + py2 + pz2 are 2 2 − − equivalent. There exist r, s Qp such that r + s = 1. (This follows, for instance, from 2 1, 1 − Corollary 1.2.15 or knowing that − − is split and using Proposition 3.3.7.) Then the Qp change of variables y ry + sz, z sy rz transforms x2 py2 pz2 to x2 + py2 + pz2, 7! 7! − − − as desired. Exercise 5.1.2. Prove the above theorem when F = Q2. 1, 1 Note that in the above proof, we showed that − − is split for p odd. In fact, we can Qp say precisely when a,b is split now. Qp Exercise 5.1.3. Suppose p is an odd rational prime and a, b Z are nonzero and square- a,b 2 free. Assume vp(a) vp(b).Show is division (i.e., ramified) if and only if Qp (1) p - a, p b,anda is a nonsquare mod p;or | 1 (2) p a, p b and a− b is a nonsquare mod p. | | − Exercise 5.1.4. Let F = Qp, p odd. Show any quadratic field extension K/F embeds in the quaternion division algebra B/F . 140 QUAINT Chapter5: Classificationofquaternionalgebras KimballMartin 5.2 Quaternion algebras over number fields In this section, let F be a number field and B denote a quaternion algebra over F . Recall that B is split at a place v of F if Bv is split, i.e., isomorphic to M2(Fv); otherwise we say B ramifies at v, or just Bv is ramified. For quaternion algebras B, B being ramified at v is equivalent to Bv being a division algebra. Note that B can only ramify at non-complex places. Let Ram(B)= v : B is ramified . { v } Proposition 5.2.1. The ramification set Ram(B) of a quaternion algebra B over a number field F is finite. Furthermore, Ram(B) determines B up to isomorphism. Proof. Write B = a,b for some a, b F . We may assume in fact that a, b .(We F 2 ⇥ 2OF may also assume that a, b are squarefree, though we do not need to for the proof.) Let S be the set primes of F which divide a or b union with the set of all even primes. Then for any finite v S, we claim Bv is split. 62 a,b For simplicity, we just show this when Fv = p, p odd. Indeed, we have Bv = where Q Qp a, b Z⇥.Ifa or b are squares, Bv is split, so assume they are not. Then, because v is 2 p odd, there are only two square classes in Zp⇥ and Kv = Qp(pa) is the unramified quadratic extension. Then, as in the proof of Theorem 5.1.1, a,b must be split because b is a norm Qp from K . v For the last part, suppose B0 is another quaternion algebra such that Ram(B0)= Ram(B). By the local classification Theorem 5.1.1, this implies B B for all v. Then v ' v0 the local-global principle Theorem 3.4.2 implies B B . ' 0 This is analogous to the fact that the ramification sets of quadratic fields Q(pd) are finite and determine the field up to isomorphism. For the next result about Ram(B), we need an important result from algebraic number theory. Theorem 5.2.2 (Quadratic Hilbert Reciprocity). Let a, b F .Thentheproductoflocal 2 ⇥ quadratic Hilbert symbols is v(a, b)Fv =1. Proof. See, e.g., [Neu99, ThmQ VI.8.1]. (This reference is a more general version than what we need, covering n-th power Hilbert symbols. Also, Neukirch’s definition of the Hilbert symbol is a little different from ours, but can be seen to be equivalent from his Proposition V.3.2 or Exercise V.3.1.) The Hilbert reciprocity law is a generalization of Gauss’s classical quadratic reciprocity. Specifically, quadratic Hilbert reciprocity can be viewed as a version of quadratic reciprocity over arbitrary number fields.1 1General Hilbert reciprocity is a law for n-th power residue symbols, but only over number fields which contain all n-th roots of unity. This generalizes other classical reciprocity laws, such as Eisenstein’s cubic reciprocity over Z[⇣3]. Hilbert reciprocity was further generalized by Artin through the development of class field theory. 141 QUAINT Chapter5: Classificationofquaternionalgebras KimballMartin Exercise 5.2.1. Deduce the classical statement of quadratic reciprocity from Hilbert reciprocity, namely: if p, q are odd (positive) rational primes, deduce p q p 1 q 1 =( 1) −2 −2 . q p − ✓ ◆✓ ◆ (Cf. Exercise 3.3.8 and Exercise 3.3.9.). Corollary 5.2.3. For a quaternion algebra B over a number field, the cardinality of Ram(B) is finite and even and contains no complex places. Proof. Recall v Ram(B) if and only if (a, b) = 1 from Corollary 3.3.8.Nowapply 2 Fv − Hilbert’s reciprocity law and Proposition 5.2.1. We now have enough results to determine if two rational quaternion algebras (i.e., quater- B = a,b B = a0,b0 nion algebras over Q) are isomorphic. Consider Q and 0 Q . We may modify a, b, a0,b0 by squares to assume that a, b, a0,b0 Z and are all squarefree.
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