A field with only a limited number of elements. It was first discovered by e Galois, so it is also called Galois domain. Compared with rational number field and real number field, it has many different properties.

catalogue

brief introduction

situation

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brief introduction

The simplest finite field is quotient ring Z/(p) obtained by modulo a prime number p of integer ring z, which is composed of p elements 0, 1, …, p- 1, and multiplied by the modulus p. ..

J Wedderburn proved in 1905 that "a finite division ring must be multiplicative commutative". So now the finite division ring is called finite field.

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situation

The set F={a, b, ...} defines two operations on the elements of f: "+"and "*", which satisfy the following three conditions.

? The elements of F 1: F form an commutative group with respect to the operation "+",and let its unit element be 0.

? The elements of F2:f \ {0} form an abelian group about the operation "*". That is, after the elements in F exclude the element 0, an exchange group is formed by the * method.

? F3: The distribution rate remains unchanged, that is, it applies to any element.

a，b，c∈F，

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a*(b+c)=(b+c)*a=a*b+a*c

When p is a prime number, it can be proved that F {0, 1, 2, …, p- 1}. In the sense of modp, the sum operation "+"and the product "*" form a domain. When the number of elements in an F field is finite, it is called a finite field.