PlanetMath: extension field

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extension field

(Definition)

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.

"extension field" is owned by drini. [ full author list (3) | owner history (2) ]

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Other names:

extension

Also defines:

degree, field extension, base field

Keywords:

fields, Galois

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Cross-references: multiplicative, projection map, polynomial, function fields, curves, algebraic, Dedekind domains, reflects, term, dimension, vector space, subfield, field
There are 186 references to this entry.

This is version 7 of extension field, born on 2001-11-08, modified 2005-02-15.
Object id is 703, canonical name is ExtensionField.
Accessed 19417 times total.

Classification:

AMS MSC:

12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous)

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

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Addition of link to Galois group and of definitions of norm and trace by gen_arboit on 2004-09-16 17:37:38

Hi,

It would be interesting to add on the possible operations on such a vector space, in the 3rd paragraph.

We know that any element of the Galois group (should have a link here) of a field extension F<E is a linear operator on E over F. Moreover, there are operations which are not field automorphisms, one of which being the multiplication by a fixed element of E. This defines the trace and norm of an element a of E in the following way.

The multiplication map A:E->E defined by Ab=ab is an F-linear operator from E to E. Consider A written in matrix form. The trace of a is defined as the trace of A, which is the sum of the eigenvalues of A. The norm of a is defined as the determinant (or norm) of A, which is the product of the eigenvalues of A. In both cases, the multiplicities are counted.

Thank you,

Genevieve Arboit

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