maths.free › Abstract Algebra › Rings › Integral Domains and Fields
Integral Domains and Fields
Let us briefly recall some definitions. If R is a commutative ring and r is a nonzero element in R, then r is said to be a zero divisor if there is some nonzero element s \in R such that rs = 0.
Integral Domains and Fields
Let us briefly recall some definitions. If \(R\) is a commutative ring and \(r\) is a nonzero element in \(R\), then \(r\) is said to be a zero divisor if there is some nonzero element \(s \in R\) such that \(rs = 0\). A commutative ring with identity is said to be an integral domain if it has no zero divisors. If an element \(a\) in a ring \(R\) with identity has a multiplicative inverse, we say that \(a\) is a unit. If every nonzero element in a ring \(R\) is a unit, then \(R\) is called a division ring. A commutative division ring is called a field.
Example
If \(i^2 = -1\), then the set \({\mathbb Z}[ i ] = \{ m + ni : m, n \in {\mathbb Z} \}\) forms a ring known as the Gaussian integers. \(\mathbb Z[i]\) the Gaussian integers It is easily seen that the Gaussian integers are a subring of the complex numbers since they are closed under addition and multiplication. Let \(\alpha = a + bi\) be a unit in \({\mathbb Z}[ i ]\). Then \(\overline{\alpha} = a - bi\) is also a unit since if \(\alpha \beta = 1\), then \(\overline{\alpha} \overline{\beta} = 1\). If \(\beta = c + di\), then \[\begin{aligned}\end{aligned}\]. Therefore, \(a^2 + b^2\) must either be \(1\) or \(-1\); or, equivalently, \(a + bi = \pm 1\) or \(a + bi = \pm i\). Therefore, units of this ring are \(\pm 1\) and \(\pm i\); hence, the Gaussian integers are not a field. We will leave it as an exercise to prove that the Gaussian integers are an integral domain.
Example
The set of matrices \[\begin{aligned}\end{aligned}\] with entries in \({\mathbb Z}_2\) forms a field.
Example
The set \({\mathbb Q}( \sqrt{2}\, ) = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \}\) is a field. The inverse of an element \(a + b \sqrt{2}\) in \({\mathbb Q}( \sqrt{2}\, )\) is \[\begin{aligned}\end{aligned}\].
We have the following alternative characterization of integral domains.
The following surprising theorem is due to Wedderburn.
For any nonnegative integer \(n\) and any element \(r\) in a ring \(R\) we write \(r + \cdots + r\) (\(n\) times) as \(nr\). We define the characteristic of a ring \(R\) to be the least positive integer \(n\) such that \(nr = 0\) for all \(r \in R\). If no such integer exists, then the characteristic of \(R\) is defined to be \(0\). We will denote the characteristic of \(R\) by \(\chr R\). \(\chr R\) characteristic of a ring \(R\)
Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.
이제 너 계산기는 이것을 해결하지 않지만, 그 조각은 계산 가능합니다. 아래의 하나를 시도하거나 자신의 것을 입력하십시오.
여기서 사용된 기호
기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.
사람들이 묻는 질문
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.
이 페이지의 일부는 다음에서 변경되었습니다. Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). 여기서 압축하고 다시 설명; 오류는 우리의.
에 더 Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula