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Concept
Integral element
Summary
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 aj 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 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.
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 ).
Integers are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q.
The Gaussian integers are the complex numbers of the form , and are integral over Z. is then the integral closure of Z in . Typically this ring is denoted .
The integral closure of Z in is the ring
This example and the previous one are examples of quadratic integers. The integral closure of a quadratic extension can be found by constructing the minimal polynomial of an arbitrary element and finding number-theoretic criterion for the polynomial to have integral coefficients. This analysis can be found in the quadratic extensions article.
Let ζ be a root of unity.
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