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A finite field is a finite set that is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms. The number of elements of a finite field is called its order or, sometimes, its size.
Every ordered field is a formally real field, i.e., 0 cannot be written as a sum of nonzero squares. [2] [3] Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order need not be uniquely determined.) The proof uses Zorn's lemma. [5]
The order of a finite field is always a prime or a power of prime. For each prime power q = p r, there exists exactly one finite field with q elements, up to isomorphism. This field is denoted GF(q) or F q. If p is prime, GF(p) is the prime field of order p; it is the field of residue classes modulo p, and its p elements are denoted 0, 1 ...
The consequences of the theorem include: the order of a group G is a power of a prime p if and only if ord(a) is some power of p for every a in G. [2] If a has infinite order, then all non-zero powers of a have infinite order as well. If a has finite order, we have the following formula for the order of the powers of a: ord(a k) = ord(a) / gcd ...
Thus a vector space of dimension d over F 1 n is a finite set of order dn on which the roots of unity act freely, together with a base point. From this point of view the finite field F q is an algebra over F 1 n, of dimension d = (q − 1)/n for any n that is a factor of q − 1 (for example n = q − 1 or n = 1).
The trace form for a finite degree field extension L/K has non-negative signature for any field ordering of K. [8] The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K. [8] If L/K is an inseparable extension, then the trace form is identically 0. [9]
The finite field with p n elements is denoted GF(p n) and is also called the Galois field of order p n, in honor of the founder of finite field theory, Évariste Galois. GF( p ), where p is a prime number, is simply the ring of integers modulo p .
In mathematics, an order in the sense of ring theory is a subring of a ring, such that . is a finite-dimensional algebra over the field of rational numbers; spans over , and; is a -lattice in .