Transcendence degree

Transcendence base is an algebraic notion from the theory of field extensions, which can be seen in analogy to the concept of vector space basis of the linear algebra. The cardinality of such a transcendence base, called the transcendence degree, represents a measure of the size of a transcendental field extension

Conceptualization

It is a field extension, ie is a subfield of the field. An element set is called known algebraically independent over when it is in the indeterminate except the zero polynomial with not a polynomial. An arbitrary subset is called algebraically independent over if every finite subset of it is. A maximal algebraically independent set in, so that you can extend by no more element to an independent algebraic over quantity, is called a transcendence basis of the field extension.

Note the analogy to linear algebra in a vector space basis can be characterized as a maximal linearly independent set.

If a field extension, then for a algebraically independent set following statements are equivalent:

• Is a transcendence basis of.
• Is algebraically, wherein the body is smallest, and contains (see Körperadjunktion ).

Transcendence degree

As the existence of a Hamel basis is proved in linear algebra, we obtain the existence of a transcendence basis by showing that any association ascending quantities algebraically independent quantities is algebraically independent again, and then use the lemma of Zorn.

If the field extension algebraically, the empty set is obviously transcendence basis. Is the field of rational functions, the field extension has transcendence basis.

In complete analogy to Steinitz Austauschlemma of linear algebra one shows that any two transcendence bases of a field extension are equally powerful. Therefore, the cardinality of a transcendence basis is an invariant of the field extension, called its transcendence degree and is denoted by. Based on the English-language name transcendence degree one finds the spelling. It follows from that is infinite, since the integer powers of a transcendent element are linearly independent over, which already has a field extension to a transcendent element, has infinite degree; the transcendence degree so does not match the degree of the field extension.

Apparently is equivalent to the statement that is algebraic. Light shows you and is made of cardinality reasons (read " beth one ", see Beth function). It has also been

• For body.

It follows immediately, the field of rational functions in indeterminates over.

Pure transcendental field extensions

A field extension is called purely transcendent, if there is a transcendence base. It follows that each element of is transcendental over. Each field extension can be used in an algebraic and split a purely transcendental field extension, as the following theorem shows:

If a field extension, so there is an intermediate body, so that the following holds

• Is purely transcendental.
• Is algebraic.

To prove this, assume for a transcendence basis over.

The augmentation of the body and are purely transcendent, wherein the non- trivial fact of transcendence of Euler's number is used for the latter. The field extension is transcendent ( ie non- algebraic ), but not purely transcendent, there is algebraic over.