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.
An element of an R-algebra is algebraic over R if it is the root of a nonzero polynomial.
A subalgebra is algebraic if all its elements are algebraic.
An algebra is algebraic if all its elements are algebraic.
A subalgebra is algebraic if and only if it is algebraic an algebra.
An algebra is algebraic if and only if it is algebraic as a subalgebra.
An integral element of an algebra is algebraic.
An element of an algebra over a field is algebraic if and only if it is integral.
If L is an algebraic field extension of K and A is an algebraic algebra over L, then A is algebraic over K.