Algebraic element

• The square root of 2 is algebraic over Q, since it is the root of the polynomial g(x) = x² − 2 whose coefficients are rational.
• Pi is transcendental over Q but algebraic over the field of real numbers R: it is the root of g(x) = x − π, whose coefficients (1 and −π) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses C/Q, not C/R.)

The following conditions are equivalent for an element ${\displaystyle a}$ of${\displaystyle L}$ :

• ${\displaystyle a}$  is algebraic over ${\displaystyle K}$ ,
• the field extension ${\displaystyle K($a)/K}   is algebraic, i.e. every element of ${\displaystyle K($a)}   is algebraic over ${\displaystyle K}$  (here ${\displaystyle K($a)}   denotes the smallest subfield of ${\displaystyle L}$  containing ${\displaystyle K}$  and ${\displaystyle a}$ ),
• the field extension ${\displaystyle K($a)/K}   has finite degree, i.e. the dimensionof${\displaystyle K($a)}   as a ${\displaystyle K}$ -vector space is finite,
• ${\displaystyle K[$a]=K(a)}  , where ${\displaystyle K[$a]}   is the set of all elements of ${\displaystyle L}$  that can be written in the form ${\displaystyle g($a)}   with a polynomial ${\displaystyle g}$  whose coefficients lie in ${\displaystyle K}$ .

To make this more explicit, consider the polynomial evaluation ${\displaystyle \varepsilon _{a}:K[$X]\rightarrow K(a),\,P\mapsto P(a)}  . This is a homomorphism and its kernelis${\displaystyle \{P\in K[$X]\mid P(a)=0\}}  . If ${\displaystyle a}$  is algebraic, this ideal contains non-zero polynomials, but as ${\displaystyle K[$X]}   is a euclidean domain, it contains a unique polynomial ${\displaystyle p}$  with minimal degree and leading coefficient ${\displaystyle 1}$ , which then also generates the ideal and must be irreducible. The polynomial ${\displaystyle p}$  is called the minimal polynomialof${\displaystyle a}$  and it encodes many important properties of ${\displaystyle a}$ . Hence the ring isomorphism ${\displaystyle K[$X]/(p)\rightarrow \mathrm {im} (\varepsilon _{a})}   obtained by the homomorphism theorem is an isomorphism of fields, where we can then observe that ${\displaystyle \mathrm {im} (\varepsilon _{a})=K($a)}  . Otherwise, ${\displaystyle \varepsilon _{a}}$  is injective and hence we obtain a field isomorphism ${\displaystyle K($X)\rightarrow K(a)}  , where ${\displaystyle K($X)}   is the field of fractionsof${\displaystyle K[$X]}  , i.e. the field of rational functionson${\displaystyle K}$ , by the universal property of the field of fractions. We can conclude that in any case, we find an isomorphism ${\displaystyle K($a)\cong K[X]/(p)}  or${\displaystyle K($a)\cong K(X)}  . 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 ${\displaystyle K}$  are again algebraic over ${\displaystyle K}$ . For if ${\displaystyle a}$  and ${\displaystyle b}$  are both algebraic, then ${\displaystyle (K($a))(b)}   is finite. As it contains the aforementioned combinations of ${\displaystyle a}$  and ${\displaystyle b}$ , adjoining one of them to ${\displaystyle K}$  also yields a finite extension, and therefore these elements are algebraic as well. Thus set of all elements of ${\displaystyle L}$  that are algebraic over ${\displaystyle K}$  is a field that sits in between ${\displaystyle L}$  and ${\displaystyle K}$ .

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 ${\displaystyle L}$  is algebraically closed, then the field of algebraic elements of ${\displaystyle L}$  over ${\displaystyle K}$  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.

• Algebraic independence

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001