An Introduction to Abstract Algebra

algebra plays a fundamental role in many areas of science and engineering. In this chapter we describe a variety of basic algebraic structures that play roles in the generation and analysis of sequences, especially sequences intended for use in communications and cryptography. This include groups (see Section 1.1), rings (see Section 1.2), and polynomials over rings (see Section 1.4). We also explore characters and Fourier transforms, basic tools for understanding structures based on groups and rings (see Section 1.3).

[1]  A. Y. Khinchin.  Continued Fractions (In Russian) , 1937 .

[2]  R. Tennant.  Algebra , 1941, Nature.

[3]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[4]  A. Weil.  On Some Exponential Sums. , 1948, Proceedings of the National Academy of Sciences of the United States of America.

[5]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[6]  R. Chien,et al.  Error-Correcting Codes, Second Edition , 1973, IEEE Transactions on Communications.

[7]  P. Deligne.  La conjecture de Weil. I , 1974 .

[8]  B. R. McDonald.  Finite Rings With Identity , 1974 .

[9]  N. Koblitz.  p-adic Numbers, p-adic Analysis, and Zeta-Functions , 1977 .

[10]  M. Newman,et al.  Topics in Algebra , 1978 .

[11]  Robert A. Scholtz,et al.  Continued fractions and Berlekamp's algorithm , 1979, IEEE Trans. Inf. Theory.

[12]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

[13]  Wolfgang M. Schmidt,et al.  Bounds for exponential sums , 1984 .

[14]  James E. Goehring,et al.  THE ENGLISH TRANSLATION , 1986 .

[15]  R. McEliece.  Finite Fields for Computer Scientists and Engineers , 1986 .

[16]  J. Chahal.  Equations over Finite Fields , 1988 .

[17]  N. Jacobson,et al.  Basic Algebra II , 1989 .

[18]  Jeffrey Shallit,et al.  Continued fractions and linear recurrences , 1993 .

[19]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[20]  Alfred J. van der Poorten,et al.  Continued Fractions of Algebraic Numbers , 1995 .

[21]  염흥렬,et al.  [서평]「Applied Cryptography」 , 1997 .

[22]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[23]  公庄 庸三.  Basic Algebra = 代数学入門 , 2002 .

[24]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[25]  Hu Chuan-Gan,et al.  On The Shift Register Sequences , 2004 .