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. Prom ...
, an element ''b'' of a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
''B'' is said to be integral over ''A'', a
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of ''B'', if there are ''n'' ≥ 1 and ''a''
''j'' in ''A'' such that
:
That is to say, ''b'' is a
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of a
monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\ ...
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
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
over" and "
algebraic extensions" in
field theory (since the root of any
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
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, "Ma ...
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
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
.
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 languag ...
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.