By Rudolf Lidl, Harald Niederreiter

**Publish 12 months note:** First released in 1994

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The concept of finite fields is a department of recent algebra that has come to the fore lately due to its various functions in such components as combinatorics, coding concept, cryptology and the mathematical learn of switching circuits.

The first a part of this booklet provides an advent to this concept, emphasizing these features which are correct for software. the second one half is dedicated to a dialogue of an important functions of finite fields, in particular to info idea, algebraic coding concept, and cryptology. there's additionally a bankruptcy on functions inside arithmetic, comparable to finite geometries, combinatorics and pseudo-random sequences.

The e-book is designed as a graduate point textbook; labored examples and copious routines that diversity from the regimen, to these giving substitute proofs of key theorems, to extensions of fabric coated within the textual content, are supplied all through.

**Read or Download Introduction to Finite Fields and Their Applications (2nd Edition) PDF**

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**Extra info for Introduction to Finite Fields and Their Applications (2nd Edition)**

**Example text**

Polynomials of degree < 0 are called constant polynomials. If R has the identity 1 and if the leading coefficient o f / ( x ) is 1, t h e n / ( x ) is called a monk polynomial. By computing the leading coefficient of the sum and the product of two polynomials, one finds the following result. 50. Theorem. Letf,ge R[x]. Then deg(/ + g) < max(deg(/),deg(g)), deg(/g)

Iii) g w f/ze momc polynomial in K[x\ of least degree having 8 as a root. Proof Property (i) was already noted and (ii) follows from the definition of g. As to (iii), it suffices to note that any monic polynomial in K[x] having 8 as a root must be a multiple of g, and so it is either equal to g or its degree is larger than that of g. D We note that both the minimal polynomial and the degree of an algebraic element 8 depend on the field K over which it is considered, so that one must be careful not to speak of the minimal polynomial or the degree of 8 without specifying A\ unless the latter is amply clear from the context.

Therefore, R/M is a field. Conversely, let R/M be a field and let J D Af, 7 =*= A/, be an ideal of /?. Then for a e 7, 0 £ Af, the residue class 0 + Af has a multi- 18 Algebraic Foundations plicative inverse, so that (a + M)(r + M) = \ + M for some r G R. This implies ar + m = 1 for some w G M. Since J is an ideal, we have 1 G J and therefore (1) = R c / , hence J = R. Thus M is a maximal ideal of P . (ii) Let P be a prime ideal of R; then R/P is a commutative ring with identity 1 + P *= 0 + P.