By John R. Pierce

ISBN-10: 0486134970

ISBN-13: 9780486134970

Covers encoding and binary digits, entropy, language and that means, effective encoding and the noisy channel, and explores ways that details idea pertains to physics, cybernetics, psychology, and artwork. "Uncommonly good...the so much enjoyable dialogue to be found." - medical American. 1980 version.

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**Example text**

Mod 97/. 14 Let a; b; m 2 Z with m 1. a; m/ divides b. mod m=d/. b; m/. 17 If a; b 2 Z with a 0 and b 3, prove that 2a C 1 is not divisible by 2b 1. 18 Prove that there is no right triangle with all side lengths being integers and such that the lengths of the two sides forming the right angle are odd. 19 For odd m 2 N, prove that the sum of the elements of every complete residue system modulo m is divisible by m. n/. n/. 22 Let m; n 2 N be such that every prime factor of m is also a prime factor of n.

It is customary to write a for the additive inverse of a 2 F and a 1 for the multiplicative inverse of a 2 F . Here are two simple properties that you use without thinking for real and complex numbers, but which hold in any field. 2 Let F be a field. Then: (i) a0 D 0 for all a 2 F; (ii) if ab D 0 for some a; b 2 F, then a D 0 or b D 0. 0 C 0/ D a0 C a0 by the distributive law, and so a0 D 0. (ii) If ab D 0 and a ¤ 0, then multiplication by a 1 yields b D a 1 0, and so b D 0 by part (i). 3 We noted that every field contains at least the two elements 0 and 1.

A characterization of multiple roots of polynomials is obtained by borrowing the concept of derivative from calculus. j aj /xj 1 2 FŒx; jD1 where j aj is the j-fold sum of aj for 1 Ä j Ä n. The usual rules for derivatives, such as the product rule, hold for every field F. 28 Let F be a field and let K be an extension field of F. x/. 25. ˛/. ˛/ D 0, and so the desired result follows. 29 Let F be a field. x// D 1 has only simple roots in every extension field of F. 28. 3 Constructions of Finite Fields Now we are ready to construct general finite fields, following in the footsteps of Evariste Galois (1811–1832).

### An Introduction to Information Theory: Symbols, Signals and Noise by John R. Pierce

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