# Prime body

The prime field is a term from the mathematical field of algebra with two different meanings. On the one hand, the smallest sub-body of a body is called its prime body; on the other hand, the term is used for finite bodies with elements, where is a prime number . Both definitions are closely related because the prime field of a field with prime number characteristics is a prime field according to the second definition. ${\ displaystyle p}$ ${\ displaystyle p}$ ## Isomorphism type of the primes

The characteristic of a field determines the isomorphism type of its prime field. Since a body is always an integrity ring , its characteristic can only be 0 or a prime number. If the characteristic is 0, the prime field is isomorphic to the field of the rational numbers . This implies that bodies whose characteristic is 0 are always infinite, after all they always contain . If, on the other hand , it is a prime number , the prime field is isomorphic to the remainder class field . But it cannot be concluded from this that fields with prime number characteristics are always finite. Infinite fields can also have finite prime fields. ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle p}$ ${\ displaystyle \ mathbb {F} _ {p}}$ ## Further properties of primes

• The prime field is the intersection of all partial fields of a body.
• Each body is the upper body of its prime body.
• It can be shown that the order of every finite field is a power of the order of its prime field.
• All primes are rigid , i.e. that is, they only have trivial automorphism . The prime body of any body can therefore be uniquely identified with one of the bodies mentioned above.

## Individual evidence

1. Martin Bossert: Channel coding. 2nd Edition. BG Teubner, Stuttgart 1998, ISBN 3-519-16143-5 , pp. 35-36