Algebraic elements and algebraic extensions #
An element of an R-algebra is algebraic over R if it is the root of a nonzero polynomial. An R-algebra is algebraic over R if and only if all its elements are algebraic over R. The main result in this file proves transitivity of algebraicity: a tower of algebraic field extensions is algebraic.
If L is an algebraic field extension of K and A is an algebraic algebra over L, then A is algebraic over K.
If A is an algebraic algebra over K, then A is algebraic over L when L is an extension of K
(a : S) / (b : S) can be reduced to
(c : S) / (d : R),
S is the integral closure of
R in an algebraic extension