Mostowski / Stark / Sneddon Introduction to Higher Algebra

1. Auflage 2014
ISBN: 978-1-4832-8035-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 37, 474 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Introduction to Higher Algebra is an 11-chapter text that covers some mathematical investigations concerning higher algebra. After an introduction to sets of functions, mathematical induction, and arbitrary numbers, this book goes on considering some combinatorial problems, complex numbers, determinants, vector spaces, and linear equations. These topics are followed by discussions of the determination of polynomials in ne variable, rings of real and complex polynomials, and algebraic and transcendental numbers. The final chapters deal with the polynomials in several variables, symmetric functions, the theory of elimination, and the quadratic and Hermitian forms. This book will be of value to mathematicians and students.

Autoren/Hrsg.

Mostowski, A.

Stark, M.

Sneddon, I. N.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Introduction to Higher Algebra;4
3;Copyright Page;5
4;Table of Contents;6
5;CHAPTER
I. INTRODUCTION;12
5.1;1. Functions;12
5.2;2. Mathematical induction;17
5.3;3. Sums and products of an arbitrary number of terms;24
6;CHAPTER
II. SOME COMBINATORIAL PROBLEMS;29
6.1;§ 1. Permutations;29
6.2;§ 2.
k-permutations;33
6.3;§ 3. Combinations;36
6.4;§ 4. Newton's multinomial formula;43
6.5;§ 5. Multiplication of permutations;46
7;CHAPTER
III. COMPLEX NUMBERS;58
7.1;§ 1. Fields;58
7.2;§ 2. Introductory remarks on complex numbers;70
7.3;§ 3. Definition of complex numbers;72
7.4;§ 4. Properties of complex numbers;81
7.5;§
5. Roots of complex numbers;94
8;CHAPTER
IV. DETERMINANTS;105
8.1;§
1. Definition of a determinant;105
8.2;§
2. Laplace expansion;115
8.3;§
3. Properties of determinants;121
8.4;§
4. Examples;128
8.5;§ 5. Cramer's formulae;140
8.6;§ 6. General Laplace theorem;146
8.7;§ 7. Cauchy's theorem and its generalizations;155
9;CHAPTER
V. VECTOR SPACES AND LINEAR EQUATIONS;164
9.1;§ 1. Vector spaces;164
9.2;§ 2. Rank of a matrix;178
9.3;§ 3. Linear equations;186
9.4;§ 4. Axiomatic definition of
determinant;199
10;CHAPTER
VI. POLYNOMIALS IN ONE VARIABLE;203
10.1;§ 1. Operations on polynomials;203
10.2;§ 2. The arithmetic of the ring
K;212
10.3;§ 3. Roots of a polynomial;235
10.4;§ 4. Interpolation formulae;243
10.5;§ 5. Rational functions;256
11;CHAPTER
VII. RINGS OF REAL AND COMPLEX POLYNOMIALS;268
11.1;§ 1. The fundamental theorem of algebra;268
11.2;§2. Polynomials of the ring B[x];277
11.3;§ 3. Quadratic equations in the domain of complex numbers;289
11.4;§ 4. Cubic equations;292
11.5;§ 5. Equations of the fourth degree;299
11.6;§ 6. Reciprocal equations;302
12;CHAPTER
VIII. RING OF RATIONAL POLYNOMIALS ALGEBRAIC AND TRANSCENDENTAL NUMBERS;306
12.1;§ 1. Reduction of polynomials with rational coefficients to polynomials
with integral coefficients;306
12.2;§ 2. Polynomials irreducible in the field W;311
12.3;§
3. Algebraic numbers;315
12.4;§
4. Transcendental numbers;322
13;CHAPTER
IX. POLYNOMIALS IN SEVERAL VARIABLES AND SYMMETRIC FUNCTIONS;329
13.1;§ 1. The arithmetic of the ring
k;329
13.2;§
2. Symmetric polynomials;345
14;CHAPTER
X. THE THEORY OF ELIMINATION;372
14.1;§
1. The resultant;372
14.2;§
2. Systems of two equations in two unknowns;380
14.3;§
3. Points of intersection of algebraic curves;386
15;CHAPTER
XI. QUADRATIC AND HERMITIAN FORMS;396
15.1;§ 1. Introduction;396
15.2;§ 2. Linear transformations;397
15.3;§ 3. Quadratic forms;415
15.4;§ 4. Orthogonal transformations of quadratic forms;432
15.5;§ 5. Hermitian forms and unitary transformations;445
16;APPENDIX:
SOME PROPERTIES OF MATRICES AND QUADRATIC FORMS;450
16.1;1. Cofactor-matrix;450
16.2;2. The case of a singular matrix;453
16.3;3. The rank of the matrix formed by minors;455
16.4;4. The rank of a symmetric matrix;456
16.5;5. Sylvester's identity. Kronecker's and Jacobi's theorems
on quadratic forms;460
16.6;6. Gram matrices;464
16.7;7. Cayley's and Hamilton's theorem;466
17;Index;470