Кеңейтімнің векторлық кеңістік ретіндегі өлшемі

Dimension of the extension field viewed as a vector space over the base field
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory—indeed in any area where fields appear prominently.

Suppose that E/F is a field extension. Then E may be considered as a vector space over F (the field of scalars). The dimension of this vector space is called the degree of the field extension, and it is denoted by [E:F]. The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. An extension E/F is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements). The degree should not be confused with the transcendence degree of a field; for example, the field Q(X) of rational functions has infinite degree over Q, but transcendence degree only equal to 1.

In this case, we start with bases uα and wβ of L/K and M/L respectively, where α is taken from an indexing set A, and β from an indexing set B. Using an entirely similar argument as the one above, we find that the products uαwβ form a basis for M/K. These are indexed by the Cartesian product A × B, which by definition has cardinality equal to the product of the cardinalities of A and B.

The complex numbers are a field extension over the real numbers with degree [C:R] = 2, and thus there are no non trivial fields between them. The field extension Q(, ), obtained by adjoining and to the field Q of rational numbers, has degree 4, that is, [Q(, ):Q] = 4. The intermediate field Q has degree 2 over Q; we conclude from the multiplicativity formula that [Q(, ):Q ] = 4/2 = 2. The finite field (Galois field) GF(125) = GF(53) has degree 3 over its subfield GF(5). More generally, if p is a prime and n, m are positive integers with n dividing m, then [GF(pm):GF(pn)] = m/n. The field extension C(T)/C, where C(T) is the field of rational functions over C, has infinite degree (indeed it is a purely transcendental extension). This can be seen by observing that the elements 1, T, T2, etc., are linearly independent over C.
The field extension C(T2) also has infinite degree over C. However, if we view C(T2) as a subfield of C(T), then in fact [C(T):C(T2)] = 2. More generally, if X and Y are algebraic curves over a field K, and F : X → Y is a surjective morphism between them of degree d, then the function fields K(X) and K(Y) are both of infinite degree over K, but the degree [K(X):K(Y)] turns out to be equal to d.

Given two division rings E and F with F contained in E and the multiplication and addition of F being the restriction of the operations in E, we can consider E as a vector space over F in two ways: having the scalars act on the left, giving a dimension [E:F]l, and having them act on the right, giving a dimension [E:F]r. The two dimensions need not agree. Both dimensions however satisfy a multiplication formula for towers of division rings; the proof above applies to left acting scalars without change.