PUMaC 2015 · 加试 · 第 6 题

Problem 6.2 (An Odd Divisor; 5 ) . Let p be a prime. Prove p divides some term of the Somos-4 sequence 2 3 { a } if and only if P = (0 , 0) has odd order in the group E ( F ) where E : y + y = x − x . n p Proof. This is a corollary of problems 4.6 and 5.2. ( ) 2 2 a a − 2 a a a a − a a n +2 n − 1 n n +1 n − 1 n +1 n − 1 n ( ⇒ ) Suppose p | a for some n ≥ 1, and then note that (2 n − 3) P = , . n 2 3 a a n n Note that the side-results of problem 5.2 show that the denominators and numbers of the x and y -coordinates ( ) 2 2 3 of (2 n − 3) P as written are coprime. Therefore, (2 n − 3) P (mod p ) = a ( a − a a ) : a a − 2 a a a : a ≡ n n − 1 n +1 n +2 n − 1 n n +1 n n − 1 n (0 : 1 : 0) (mod p ). Therefore, note an odd multiple of P equals the identity; thus P has odd order. k +3 ( ⇐ ) Secondly if kP = (0 : 1 : 0) (mod p ) where k is odd, then (2 · ( ) − 3) P = (0 : 1 : 0), and p | a . ( k +3) / 2 2 This is a rather magical connection between divisibility of a sequence and elliptic curves, don’t you think? However, strangely enough, we will soon be able to make even weirder equivalent statements. 24 PUMaC 2015 Power Round Section 7 page 25 7 Galois Theory Unfortunately, for the sake of time (we can’t build up all of Galois theory from scratch for this Power Round!), we won’t be able to give more than a heuristic of some of the methods we use here. (Un)Luckily for you, the reader, this also means there aren’t many problems directly on Galois theory :(. We hope that regardless of your mathematical background, this section is still interesting enough to try to understand. We describe first some of the necessary groundwork. We previously introduced a fundamental object of algebra: groups. This was essentially the most basic “thing” we could do math on. We have only one operation on a group at all times. Anything simpler would have very little to it. Since then, we further saw a taste of more complicated algebraic objects, which as promised, we explore here. One step up from the group is another essential object of mathematics: a ring. Rings in some sense can be thought of as an extension of (additive) groups. Definition 27. Let R be a set of elements that has a closed, binary operation we call addition such that { R, + } is an additive, commutative group and a second closed, binary, commutative operation · that we can call multiplication. Suppose R has these properties: • The operation · is associative. • For all a, b, c ∈ R , a · ( b + c ) = a · b + a · c, and ( b + c ) · a = b · a + c · a. • Finally, there exists a multiplicative identity we call 1 such that for all a ∈ R , a · 1 = 1 · a = a . Then R is called a ring. The consequences of such a definition is that R contains 0 (necessary by the addition law), and 1 (necessary by the multiplication law). Many definitions also add that 1 6 = 0, but this is not strictly necessary.