Subfield

Definition

Let $L$ be a field. A subfield of $L$ is a subring $K \subseteq L$ such that $a^{-1} \in K$ for every nonzero $a \in K$ .

Remarks

Equivalently, $K$ contains $0$ and $1$ , and $K$ is closed under addition, negation, multiplication and the inverse of nonzero elements. A subfield is a field with the operations of $L$ restricted to $K$ . The rationals $\mathbb{Q}$ are a subfield of $\mathbb{R}$ , and $\mathbb{R}$ is a subfield of $\mathbb{C}$ . The intersection of any family of subfields of $L$ is a subfield of $L$ . The intersection of all of them is the prime subfield.

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Group concept
Ring definition
Commutative ring definition
Subring definition
Unit of a ring definition
Field definition
Subfield definition

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115 pages build on this one.

Tag	`07QY5RPFBR`
URI	`m2://07QY5RPFBR`
Kind	definition
Writing	W0 stub
MSC	12F99
Defines	Subfield
Uses	Field, Subring

Subfield

Definition

Remarks

Reading path

Where this leads

Further reading

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