Vipul: Created page with '{{field extension property}} ==Definition== Suppose $L$ is a field extension of a field $K$ . We say that $L$ is '''purely transcendenta...'

2009-05-10T16:44:10Z

Created page with '{{field extension property}} ==Definition== Suppose <math>L</math> is a field extension of a field <math>K</math>. We say that <math>L</math> is '''purely transcendenta...'

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{{field extension property}}

==Definition==

Suppose <math>L</math> is a [[field extension]] of a [[field]] <math>K</math>. We say that <math>L</math> is '''purely transcendental''' if there exists a subset <math>S</math> of <math>L</math> such that <math>S</math> generates <math>L</math> over <math>K</math> and <math>S</math> is algebraically independent over <math>K</math>: in other words, for any polynomial <math>p \in K[x_1,x_2, \dots, x_n]</math>, and any distinct <math>s_1,s_2, \dots, s_n \in S</math>, <math>p(s_1,s_2, \dots, s_n) \ne 0</math>.

Such a subset <math>S</math> is termed a '''transcendence base'''.

==Relation with other properties==

===Opposite properties===

* [[Algebraic extension]]

==Facts==

* [[Steinitz theorem]] states that every field extension is an [[algebraic extension]] of a [[purely transcendental extension]].
* [[Luroth's theorem]] states that any sub-extension of a purely transcendental extension with transendence base of size one is also purely transcendental.

Purely transcendental extension - Revision history

Vipul: Created page with '{{field extension property}} ==Definition== Suppose L is a field extension of a field K. We say that L is '''purely transcendenta...'

Vipul: Created page with '{{field extension property}} ==Definition== Suppose $L$ is a field extension of a field $K$ . We say that $L$ is '''purely transcendenta...'