algebraicAlgebraic2002-01-05 01:23:332005-03-15 06:07:36Definitiontranscendental\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}Let $B$ be a ring with a subring $A$. An element $x \in B$ is {\em algebraic} over $A$ if there exist elements $a_1, \dots, a_n \in A$, with $a_n \neq 0$, such that
$$
a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0.
$$
An element $x \in B$ is {\em transcendental} over $A$ if it is not algebraic.
The ring $B$ is {\em algebraic} over $A$ if every element of $B$ is algebraic over $A$.