|
|
|
Viewing Version
4
of
'extension field'
|
[ view 'extension field'
|
back to history
]
| Title of object: |
extension field |
| Canonical Name: |
ExtensionField |
| Type: |
Definition |
| Created on: |
2001-11-08 00:49:41 |
| Modified on: |
2002-05-05 01:42:44 |
| Classification: |
msc:12F99 |
| Keywords: |
fields, Galois |
| Defines: |
degree |
| Synonyms: |
extension field=extension |
Revision comment (for changes between this and next version):
| Changes for correction #4537 ('Also define `field extension''). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage[all]{xypic} |
Content:
We say that a field $K$ is an \emph{extension} of $F$ if $F$ is a subfield of $K$.
We usually denote $K$ being an extension of $F$ by: $F \subset K$, $F\le K$, $K/F$, or
\xymatrix{
K \ar@{-}[d] \\
If $K$ is an extension of $F$, we can regard $K$ as a vector space over $F$. The dimension of this space (which could possibly be infinite) is denoted $[K:F]$, and called the {\em degree} of the extension\footnote{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.}.
One of the classic theorems on extensions states that if $F\subset K\subset L$, then
$$[L:F]=[L:K][K:F]$$
(in other words, degrees are multiplicative in towers). |
|
|
|
|
|
