# euclidean domains

Published articles for euclidean domains.

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## (Finite) Fields -- A Primer

DevFeed: [(Finite) Fields -- A Primer](<https://devfeed.tech/articles/finite-fields-a-primer-40348.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2014/02/26/finite-fields-a-primer/>)

Published: 2014-02-26T10:00:01Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>)

Tags: [complex-numbers](<https://devfeed.tech/tags/complex-numbers.md>), [cryptography](<https://devfeed.tech/tags/cryptography.md>), [euclidean-domains](<https://devfeed.tech/tags/euclidean-domains.md>), [field](<https://devfeed.tech/tags/field.md>), [field-characteristic](<https://devfeed.tech/tags/field-characteristic.md>), [finite-fields](<https://devfeed.tech/tags/finite-fields.md>), [groups](<https://devfeed.tech/tags/groups.md>), [ideals](<https://devfeed.tech/tags/ideals.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [monoids](<https://devfeed.tech/tags/monoids.md>), [operations](<https://devfeed.tech/tags/operations.md>), [polynomial-ring](<https://devfeed.tech/tags/polynomial-ring.md>), [vector-spaces](<https://devfeed.tech/tags/vector-spaces.md>)

### AI overview

This primer introduces fields as commutative rings with 0 and 1 in which every nonzero element has a multiplicative inverse. It defines the field axioms, places fields within related algebraic structures, and raises the question of finite fields.

### Source excerpt

So far on this blog we've given some introductory notes on a few kinds of algebraic structures in mathematics (most notably groups and rings, but also monoids). Fields are the next natural step in the progression. If the reader is comfortable with rings, then a field is extremely simple to describe: they're just commutative rings with 0 and 1, where every nonzero element has a multiplicative inverse. We'll give a list of all of the properties that go into this "simple" definition in a moment, but an even more simple way to describe a field is as a place where "arithmetic makes sense.