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Curve Algebraic Vector Space Euclidean Transformation Dimension Field Basic Polynomial Function Field Subfield Multiplicative Function Generalized Cartesian Product Dedekind Domain

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Extension Field

We say that a field is an extension of if is a subfield of .

We usually denote being an extension of by $F\subset K$ , $F\le K$ , or

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ K \ar@{-}[d] \\ F } } \end{xy}$

One may speak of the field extension

and call

the base field.

If is an extension of , we can regard as a vector space over . The dimension of this space (which could possibly be infinite) is denoted , and called the degree of the extension.¹

One of the classic theorems on extensions states that if $F\subset K\subset L$ , then

$\displaystyle [L:F]=[L:K][K:F]$

(in other words, degrees are multiplicative in towers).

Footnotes

... extension.¹: The term “degree” reflects the fact that, in the more general setting of Dedekind domains and scheme-theoretic algebraic curves, the degree of an extension of function fields equals the algebraic degree of the polynomial defining the projection map of the underlying curves.

1	Curve
1	Euclidean Transformation
1	Vector Space
1	Dimension
1	Basic Polynomial
1	Field
1	Algebraic
2	Subfield
2	Generalized Cartesian Product
2	Multiplicative Function
2	Function Field
3	Dedekind Domain