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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.

Symbols used here

\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
i
imaginary unit
i² = −1.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
(G, \cdot),\ e,\ g^{-1}
group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.
G \cong H,\ G / N
isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
\operatorname{Hom}(A, B),\ f \circ g
arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

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.

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Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

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