Degree of a field extension

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Degree of a field extension: If E is a field containing F, then E is a vector space over F. The dimension of this vector space is called the degree of the extension and is written [E:F]. This degree can be finite or infinite: a finite degree means E/F is a finite extension; otherwise it is an infinite extension. The degree measures how big E is relative to F, but it is not the same as the size of the field (fields can be infinite). It also differs from transcendence degree; for example, the field Q(X) has infinite degree over Q but transcendence degree 1.

Tower property: If K ⊆ L ⊆ M are fields, then [M:K] = [M:L] · [L:K]. This holds for both finite and infinite degrees (in the infinite case, interpret the product as a product of cardinals). In particular, if [M:K] is finite, then both [M:L] and [L:K] are finite.

Consequences: If [M:K] = p is a prime number, there are no intermediate fields between K and M—every intermediate field is either K or M. For example, Q ⊆ Q(√2) has degree 2, and there are no fields strictly between them.

How the multiplication works: Suppose [L:K] = d and [M:L] = e are finite. Take a basis u1, ..., ud for L over K and a basis w1, ..., we for M over L. Then the products u_i w_j (i = 1..d, j = 1..e) form a basis for M over K, giving [M:K] = de. The same idea works with bases indexed by sets A and B, giving a basis uα wβ for M over K indexed by A × B, whose size is |A|·|B|.

Left vs right dimensions: If E and F are division rings with F ⊆ E, you can view E as a vector space over F on the left or on the right. The left and right dimensions [E:F]_l and [E:F]_r may differ, but the tower multiplicativity still holds for the left action.