---
tag: 0MA5B744K3
kind: definition
name: Field generated by adjoining elements
msc: [12F05]
defines: 05NAS7AGMZ
uses: [0WQ1A8SHTZ, 07QY5RPFBR, 02NBKVQFRQ]
sources:
  - url: https://en.wikipedia.org/wiki/Field_extension
    locator: Extension field
---

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $S \subseteq L$.
The field $K(S)$ is the intersection of all the [[subfields|07QY5RPFBR]] of $L$ that contain $K$ and $S$.
It is the smallest [[intermediate field|02NBKVQFRQ]] of $L/K$ that contains $S$.

## Remarks

For $S = \{a_1, \dots, a_n\}$ one writes $K(a_1, \dots, a_n)$.
The elements of $K(S)$ are the quotients $f(s_1, \dots, s_n)/g(s_1, \dots, s_n)$ with $f, g$ polynomials over $K$, $s_i \in S$ and $g(s_1, \dots, s_n) \neq 0$.
One has $K(a, b) = K(a)(b)$.
For example, $\mathbb{Q}(\sqrt{2}) = \{ p + q \sqrt{2} : p, q \in \mathbb{Q} \}$ inside $\mathbb{R}$.

## Explanations

### L1 human

To adjoin elements to a field, you take the smallest field that contains the old field and the new elements.
It contains everything you can get from them by addition, subtraction, multiplication and division.