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Concept
Degree of a field extension
Summary
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.
Given three fields arranged in a tower, say K a subfield of L which is in turn a subfield of M, there is a simple relation between the degrees of the three extensions L/K, M/L and M/K:
In other words, the degree going from the "bottom" to the "top" field is just the product of the degrees going from the "bottom" to the "middle" and then from the "middle" to the "top". It is quite analogous to Lagrange's theorem in group theory, which relates the order of a group to the order and index of a subgroup — indeed Galois theory shows that this analogy is more than just a coincidence.
The formula holds for both finite and infinite degree extensions. In the infinite case, the product is interpreted in the sense of products of cardinal numbers. In particular, this means that if M/K is finite, then both M/L and L/K are finite.
If M/K is finite, then the formula imposes strong restrictions on the kinds of fields that can occur between M and K, via simple arithmetical considerations.
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