> For the complete documentation index, see [llms.txt](https://cryptohack.gitbook.io/cryptobook/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://cryptohack.gitbook.io/cryptobook/abstract-algebra/fields.md).

# Fields

A set$$F$$with two binary operations $$+,\cdot:F\times F\to F$$is a **field** if the following holds:

* $$R,+$$is a commutative group with identity $$0$$
* $$R-{0},\cdot$$is a commutative group with identity$$1$$.
* Distributivity: $$a(b+c)=ab+ac,(a+b)c=ac+bc$$

// field extensions, algebraic elements