By Dickson, Leonard Eugene

This in-depth advent to classical themes in larger algebra presents rigorous, exact proofs for its explorations of a few of arithmetic' most vital thoughts, together with matrices, invariants, and teams. Algebraic Theories experiences the entire vital theories; its large choices variety from the rules of upper algebra and the Galois thought of algebraic equations to finite linear groups Read more...

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P does not have the implicit factor a 0). For, if so, then P' == P (0 , S2, S 3, d) = 0, d = 4ai3 a3 — 3ai2 a22. Since a3 occurs in d, but in neither S 2 nor 2 3, we conclude that P 9 is free of d. The first power of 23 = 2ai3 is not cancelled by a poly nomial in S 2 = — ai2. Hence P 9 is free also of 23 and therefore of S2. Hence every seminvariant of the cubic form / is a polynomial in do, S2, S3, D. , a rela tion not serving to express any one of the four as a polynomial in the remaining three.

RANK OF MATRIX §26] 49 26. Rank of a matrix. Every matrix M having more than one element contains other matrices obtained from M by deleting cer tain rows or columns or both. In particular, it contains certain square matrices. The determinants of these square matrices are called the determinants of M. A matrix is said to be of rank r if it contains at least one r-rowed determinant which is not zero, while all its determinants of order higher than r are zero. The zero matrix all of whose elements are zero is said to be of rank 0.

J t be an arrangement of gi,. . ,gt which is derived from gi, . . , gt by l successive interchanges of two terms. Hence A may be derived from a by Z successive interchanges of two columns, so that A = ( — l ) 1a. The sum of the n\ products ( — 1)* 6 , ^ •••bjtkt 1, . . , n so that gi < g2 < ■■■<9 §27] BILINEAR FORMS 51 corresponding to all such arrangements ji, . . , j t is, by definition, the expansion of the determinant ^1*1 bgxkt ' kgtkt which is a determinant of B. Hence D = 2Z a & where the sum mation extends over all the selections gi, .