Editing Algebraic element

{{Short description|Concept in abstract algebra}}
In [[mathematics]], if {{math|''A''}} is an [[associative algebra]] over {{math|''K''}}, then an element {{math|''a''}} of {{math|''A''}} is an '''algebraic element''' over {{math|''K''}}, or just '''algebraic over''' {{math|''K''}}, if there exists some non-zero [[polynomial]] <math>g(x) \in K[x]</math> with [[coefficient]]s in {{math|''K''}} such that {{math|''g''(''a'') {{=}} 0}}.<ref>{{Cite book |last=Roman |first=Steven |title=Advanced Linear Algebra |date=2008 |publisher=Springer New York Springer e-books |isbn=978-0-387-72831-5 |series=Graduate Texts in Mathematics |location=New York, NY |pages=458–459 |chapter=18}}</ref> Elements of {{math|''A''}} that are not algebraic over {{math|''K''}} are '''transcendental over''' {{math|''K''}}. A special case of an associative algebra over <math>K</math> is an [[extension field]] <math>L</math> of <math>K</math>.

These notions generalize the [[algebraic number]]s and the [[transcendental number]]s (where the field extension is {{math|'''C'''/'''Q'''}}, with {{math|'''C'''}} being the field of [[complex number]]s and {{math|'''Q'''}} being the field of [[rational number]]s).

== Examples ==
* The [[square root of 2]] is algebraic over {{math|'''Q'''}}, since it is the root of the polynomial {{math|''g''(''x'') {{=}} ''x''<sup>2</sup> − 2}} whose coefficients are rational.
* [[Pi]] is transcendental over {{math|'''Q'''}} but algebraic over the field of [[real number]]s {{math|'''R'''}}: it is the root of {{math|''g''(''x'') {{=}} ''x'' − π}}, whose coefficients (1 and −{{pi}}) are both real, but not of any polynomial with only rational coefficients.  (The definition of the term [[transcendental number]] uses {{math|'''C'''/'''Q'''}}, not {{math|'''C'''/'''R'''}}.)

== Properties ==

The following conditions are equivalent for an element <math>a</math> of an extension field <math>L</math> of <math>K</math>:
* <math>a</math> is algebraic over <math>K</math>,
* the field extension <math>K(a)/K</math> is algebraic, i.e. ''every'' element of <math>K(a)</math> is algebraic over <math>K</math> (here <math>K(a)</math> denotes the smallest subfield of <math>L</math> containing <math>K</math> and <math>a</math>),
* the field extension <math>K(a)/K</math> has finite degree, i.e. the [[dimension of a vector space|dimension]] of <math>K(a)</math> as a <math>K</math>-[[vector space]] is finite,
* <math>K[a] = K(a)</math>, where <math>K[a]</math> is the set of all elements of <math>L</math> that can be written in the form <math>g(a)</math> with a polynomial <math>g</math> whose coefficients lie in <math>K</math>.

To make this more explicit, consider the [[Polynomial_ring#Polynomial_evaluation|polynomial evaluation]] <math>\varepsilon_a: K[X] \rightarrow K(a),\, P \mapsto P(a)</math>. This is a [[homomorphism]] and its [[Homomorphism#Kernel|kernel]] is <math>\{P \in K[X] \mid P(a) = 0 \}</math>. If <math>a</math> is algebraic, this [[Ideal_(ring_theory)|ideal]] contains non-zero polynomials, but as <math>K[X]</math> is a [[euclidean domain]], it contains a unique polynomial <math>p</math> with minimal degree and leading coefficient <math>1</math>, which then also generates the ideal and must be [[Irreducible_polynomial|irreducible]]. The polynomial <math>p</math> is called the [[Minimal polynomial (field theory)|minimal polynomial]] of <math>a</math> and it encodes many important properties of <math>a</math>. Hence the ring isomorphism <math>K[X]/(p) \rightarrow \mathrm{im}(\varepsilon_a)</math> obtained by the [[Isomorphism_theorems#Theorem_A_(rings)|homomorphism theorem]] is an isomorphism of fields, where we can then observe that <math>\mathrm{im}(\varepsilon_a) = K(a)</math>. Otherwise, <math>\varepsilon_a</math> is injective and hence we obtain a field isomorphism <math>K(X) \rightarrow K(a)</math>, where <math>K(X)</math> is the [[field of fractions]] of <math>K[X]</math>, i.e. the [[Rational_function#Abstract_algebra_and_geometric_notion|field of rational functions]] on <math>K</math>, by the universal property of the field of fractions. We can conclude that in any case, we find an isomorphism <math>K(a) \cong K[X]/(p)</math> or <math>K(a) \cong K(X)</math>. Investigating this construction yields the desired results.

This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over <math>K</math> are again algebraic over <math>K</math>. For if <math>a</math> and <math>b</math> are both algebraic, then <math>(K(a))(b)</math> is finite. As it contains the aforementioned combinations of <math>a</math> and <math>b</math>, adjoining one of them to <math>K</math> also yields a finite extension, and therefore these elements are algebraic as well. Thus set of all elements of <math>L</math> that are algebraic over <math>K</math> is a field that sits in between <math>L</math> and <math>K</math>. 

Fields that do not allow any algebraic elements over them (except their own elements) are called [[algebraically closed field|algebraically closed]]. The field of complex numbers is an example. If <math>L</math> is algebraically closed, then the field of algebraic elements of <math>L</math> over <math>K</math> is algebraically closed, which can again be directly shown using the characterisation of simple algebraic extensions above. An example for this is the [[Algebraic number|field of algebraic numbers]].

==See also==
*[[Algebraic independence]]

==References==
{{reflist}}

== Further reading ==
*{{Lang Algebra | edition=3r}}

[[Category:Algebraic properties of elements]]