What does algebraic extension mean?
Definitions for algebraic extension
al·ge·bra·ic ex·ten·sion
This dictionary definitions page includes all the possible meanings, example usage and translations of the word algebraic extension.
Wikipedia
Algebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental. For example, the field extension R/Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C/R and Q(√2)/Q are algebraic, where C is the field of complex numbers. All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic. The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers. If a is algebraic over K, then K[a], the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. The converse is true as well, if K[a] is a field, then a is algebraic over K. In the special case where K = Q is the field of rational numbers, Q[a] is an example of an algebraic number field. A field with no proper algebraic extensions is called algebraically closed. An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice. An extension L/K is algebraic if and only if every sub K-algebra of L is a field.
Numerology
Chaldean Numerology
The numerical value of algebraic extension in Chaldean Numerology is: 7
Pythagorean Numerology
The numerical value of algebraic extension in Pythagorean Numerology is: 3
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"algebraic extension." Definitions.net. STANDS4 LLC, 2024. Web. 30 Apr. 2024. <https://www.definitions.net/definition/algebraic+extension>.
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