Definitions for integral element
in·te·gral el·e·ment

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1. Integral element

   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 b n + a n − 1 b n − 1 + ⋯ + a 1 b + a 0 = 0. {\displaystyle b^{n}+a_{n-1}b^{n-1}+\cdots +a_{1}b+a_{0}=0.} 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., 2 {\displaystyle {\sqrt {2}}} or 1 + i {\displaystyle 1+i} ); 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.

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1. Integral element

   In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there is an 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. The special case of greatest interest in number theory is that of complex numbers integral over Z; in this context, they are usually called algebraic integers. A ring consisting of the algebraic integers of a finite extension field k of the rationals Q is called the ring of integers of k, a central object in algebraic number theory. 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. In this article, the term ring will be understood to mean commutative ring with a unity.

How to pronounce integral element?

1. Alex

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   Mark

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   Daniel

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   Libby

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Numerology

1. Chaldean Numerology

   The numerical value of integral element in Chaldean Numerology is: 1

2. Pythagorean Numerology

   The numerical value of integral element in Pythagorean Numerology is: 7

References

1. ^ Wikipedia
   https://en.wikipedia.org/wiki/Integral_Element
2. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=integral element