In
commutative algebra, an element
b of a
commutative ring B is said to be
integral over A, a
subring of
B, if there are
n = 1 and
such that
That is to say,
b is a root of a
monic polynomial over
A. If every element of
B is integral over
A, then it is said that
B is
integral over A, or equivalently
B is an
integral extension of
A. If
A,
B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "
algebraic extensions" in
field theory (since the root of any polynomial is the root of a monic polynomial).
