Template:Short description In mathematics, if Template:Math is an associative algebra over Template:Math, then an element Template:Math of Template:Math is an algebraic element over Template:Math, or just algebraic over Template:Math, if there exists some non-zero polynomial <math>g(x) \in K[x]</math> with coefficients in Template:Math such that Template:Math.<ref>Template:Cite book</ref> Elements of Template:Math that are not algebraic over Template:Math are transcendental over Template:Math. 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 numbers and the transcendental numbers (where the field extension is Template:Math, with Template:Math being the field of complex numbers and Template:Math being the field of rational numbers).
ExamplesEdit
- The square root of 2 is algebraic over Template:Math, since it is the root of the polynomial Template:Math whose coefficients are rational.
- Pi is transcendental over Template:Math but algebraic over the field of real numbers Template:Math: it is the root of Template:Math, whose coefficients (1 and −Template:Pi) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses Template:Math, not Template:Math.)
PropertiesEdit
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 <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 evaluation <math>\varepsilon_a: K[X] \rightarrow K(a),\, P \mapsto P(a)</math>. This is a homomorphism and its kernel is <math>\{P \in K[X] \mid P(a) = 0 \}</math>. If <math>a</math> is algebraic, this 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. The polynomial <math>p</math> is called the 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 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 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. 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 field of algebraic numbers.