Algebraic theories by Dickson, Leonard Eugene

By Dickson, Leonard Eugene

This in-depth creation to classical issues in larger algebra offers rigorous, particular proofs for its explorations of a few of arithmetic' most vital suggestions, together with matrices, invariants, and teams. Algebraic Theories experiences the entire vital theories; its huge choices diversity from the rules of upper algebra and the Galois idea of algebraic equations to finite linear groups  Read more...

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This contra­ diction completes the proof of Theorem 12. C o r o l l a r y . The weight of an invariant of degree d of a binary form of order p is \pd. 11. Theorem 13. Every binary linear transformation is a product of transformations of the three types Tn: Sm: x = £ + nrj, x = £, y = v; y = my (m V: x = -v , y =Z- 0); From these we obtain Vz = V~l: x = 77, y = — £, and 1That one such K always exists is proved in §16. § 11] GENERATORS OF TRANSFORMATIONS Rn = V- 1 T . n V: Pm = V~l SmV: 17 x = X, y = Y + n X ; x = mX, y = Y (m * 0).

Ch. I ALGEBRAIC INVARIANTS 16 covariants by Theorem 10. Our next problem will then be to find all covariants K having a given leader S. As an aid to its solution, we now prove T h e o r e m 1 2 . 1 By Theorem 9, the weight w of S is the index l of K. By Theorem 7 with r = 1, q = 2, we have pd — n = 21. Hence the order n of K is uniquely determined by S. If there exist two distinct covariants S x n + •••of /, they have the same index l = w, whence their difference is a covariant of / having the factor y.

Since OQS has the same degree and weight as S, we may employ the formula derived from this by replacing r by r — 1 to eliminate Or— 2 £2r_1, and by repetitions of this process evidently obtain 0 T~l QrS = i2[OQ - (n + 2 )][0 0 - 2{n + 3)] . . [ O i 2 - ( r - l ) ( n + r)]S, which we may also establish by induction. Let S be of degree d and weight w = §pd, so that its n in (8) is zero. Take r = w + 1, 33 FINITENESS OF COVARIANTS §18] apply i2w+1 S = 0, and divide by ( — 2) ( — 2 •3 ) . . X (14) OQ00 ^ OQnr r U ii “i » - [ !

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