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 dimension of ${\displaystyle K(a)}$  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 kernel is ${\displaystyle \{P\in K[X]\mid P(a)=0\}}$ . 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 polynomial of ${\displaystyle a}$  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 fractions of ${\displaystyle K[X]}$ , i.e. the field of rational functions on ${\displaystyle K}$ , 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