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In mathematics, a finite field is a field that contains a finite number of elements. In other words, a finite field is a finite set on which the four basic operations – addition, subtraction, multiplication and division (excluding division by zero) – are defined and satisfy the field axioms/rules of the arithmetic. Finite fields are an important area of mathematics and computer science and are widely used in geometry, finite geometry, algebraic geometry, number theory, coding theory and cryptography.
A finite field is also known as Galois field.
Any finite field must have a characteristic which is not zero, since containing a characteristic which is zero would lead it to be infinite. In a finite field, the number of elements is known as its order. The order of any finite field needs to be the power of a prime. In other words, Any finite field has pk elements where p is a prime and k a positive integer. In a finite field, a field comprised of no proper sub-fields is known as a prime field. All fields of a given order are isomorphic.