Algebra Trigonometry

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By I. Kaplansky

An algebraic prelude Continuity of automorphisms and derivations $C^*$-algebra axiomatics and uncomplicated effects Derivations of $C^*$-algebras Homogeneous $C^*$-algebras CCR-algebras $W^*$ and $AW^*$-algebras Miscellany Mappings keeping invertible components Nonassociativity Bibliography

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If p1 , p2 , p3 ∈ X are the three intersection points (counted with multiplicity) of a line and X, then p1 + p2 + p3 = 0. 48 5 The group structure on an elliptic curve Let us show how this determines the group structure. If p1 and p2 are two points on X, then we draw a line through them (or the tangent to X at them if they are equal) and let p3 be the third intersection point. , p3 = −(p1 + p2 ) so that in order to get the sum of p1 + p2 we need to find the inverse of p3 . , the line given by its x-coordinate being equal to that of p3 ) and let p4 be the third point of intersection.

We shall see a simpler proof of this fact later. What does this relation mean? Well, we have the following linear algebra exercise. Exercise 20. Show that the three pairs (r, s), (u, v), and (x, y) of C2 (or R2 ) lie on a line precisely when r s 1 u v 1 = 0. x y 1 Hence what the relation says is that if u + v + w = 0, then (℘ (u), ℘ (u)), (℘ (v), ℘ (v)), and (℘ (w), ℘ (w)) lie on a line. Now, u, v, w is the value of the elliptic integral and we know by the addition formula that u + v + w = 0 if (℘ (u), ℘ (u)), (℘ (v), ℘ (v)), and (℘ (w), ℘ (w)) lie on a line.

The homogeneity implies that the condition F (s, t) = 0 depends only on the homogeneous coordinate (s : t) (as it should). By construction the zeroes of F , seen as points on P1 (k) and then as points on , are the points of intersection between and X but we may also attach a multiplicity to each such point. Exercise 32. i) Show that every homogeneous polynomial F (s, t) = 0 over an algebraically closed field factors as i (bi s − ai t)ni , where the points (ai : bi ) are the distinct zeroes of F and the multiplicities ni are uniquely determined by the zero (ai : bi ).

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