In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
, 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 ''a''
''j'' in ''A'' such that
:
That is to say, ''b'' is a
root of a
monic polynomial over ''A''. The set of elements of ''B'' that are integral over ''A'' is called the integral closure of ''A'' in ''B''. It is a subring of ''B'' containing ''A''. If every element of ''B'' is integral over ''A'', then we say 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).
The case of greatest interest in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
is that of
complex numbers integral over Z (e.g.,
or
); in this context, the integral elements are usually called
algebraic integers. The algebraic integers in a finite
extension field ''k'' of the
rationals Q form a subring of ''k'', called the
ring of integers of ''k'', a central object of study in
algebraic number theory.
In this article, the term ''
ring'' will be understood to mean ''commutative ring'' with a multiplicative identity.
Examples
Integral closure in algebraic number theory
There are many examples of integral closure which can be found in algebraic number theory since it is fundamental for defining the
ring of integers for an
algebraic field extension (or
).
Integral closure of integers in rationals
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q.
Quadratic extensions
The
Gaussian integers are the complex numbers of the form
, and are integral over Z.