Example 9.3.1 (Rational numbers). The rational numbers form a field. It is called the *field of rational numbers* and denoted $\mathbf{Q}$.

## 9.3 Examples of fields

To get started, let us begin by providing several examples of fields. The reader should recall that if $R$ is a ring and $I \subset R$ an ideal, then $R/I$ is a field precisely when $I$ is a maximal ideal.

Example 9.3.2 (Prime fields). If $p$ is a prime number, then $\mathbf{Z}/(p)$ is a field, denoted $\mathbf{F}_ p$. Indeed, $(p)$ is a maximal ideal in $\mathbf{Z}$. Thus, fields may be finite: $\mathbf{F}_ p$ contains $p$ elements.

Example 9.3.3. In a principal ideal domain, an ideal generated by an irreducible element is maximal. Now, if $k$ is a field, then the polynomial ring $k[x]$ is a principal ideal domain. It follows that if $P \in k[x]$ is an irreducible polynomial (that is, a nonconstant polynomial that does not admit a factorization into terms of smaller degrees), then $k[x]/(P)$ is a field. It contains a copy of $k$ in a natural way. This is a very general way of constructing fields. For instance, the complex numbers $\mathbf{C}$ can be constructed as $\mathbf{R}[x]/(x^2 + 1)$.

Example 9.3.4 (Quotient fields). Recall that, given a domain $A$, there is an imbedding $A \to F$ into a field $F$ constructed from $A$ in exactly the same manner that $\mathbf{Q}$ is constructed from $\mathbf{Z}$. Formally the elements of $F$ are (equivalence classes of) fractions $a/b$, $a, b \in A$, $b \not= 0$. As usual $a/b = a'/b'$ if and only if $ab' = ba'$. The field $F$ is called the *quotient field*, or *field of fractions*, or *fraction field* of $A$. The quotient field has the following universal property: given an injective ring map $\varphi : A \to K$ to a field $K$, there is a unique map $\psi : F \to K$ making

commute. Indeed, it is clear how to define such a map: we set $\psi (a/b) = \varphi (a)\varphi (b)^{-1}$ where injectivity of $\varphi $ assures that $\varphi (b) \not= 0$ if $ b \not= 0$.

Example 9.3.5 (Field of rational functions). If $k$ is a field, then we can consider the field $k(x)$ of rational functions over $k$. This is the quotient field of the polynomial ring $k[x]$. In other words, it is the set of quotients $F/G$ for $F, G \in k[x]$, $G \not= 0$ with the obvious equivalence relation.

Example 9.3.6. Let $X$ be a Riemann surface. Let $\mathbf{C}(X)$ denote the set of meromorphic functions on $X$. Then $\mathbf{C}(X)$ is a ring under multiplication and addition of functions. It turns out that in fact $\mathbf{C}(X)$ is a field. Namely, if a nonzero function $f(z)$ is meromorphic, so is $1/f(z)$. For example, let $S^2$ be the Riemann sphere; then we know from complex analysis that the ring of meromorphic functions $\mathbf{C}(S^2)$ is the field of rational functions $\mathbf{C}(z)$.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)