Editing Characteristic (algebra) (section)

== Case of fields <span class="anchor" id="Fields"></span> ==
<!-- This section is an {{R to anchor}} from [[Characteristic exponent of a field]] and [[Prime characteristic]] -->
As mentioned above, the characteristic of any [[Field (mathematics)|field]] is either {{math|0}} or a prime number. A field of non-zero characteristic is called a field of ''finite characteristic'' or ''positive characteristic'' or ''prime characteristic''. The ''characteristic exponent'' is defined similarly, except that it is equal to {{math|1}} when the characteristic is {{math|0}}; otherwise it has the same value as the characteristic.<ref>
{{cite book
 | last = Bourbaki | first = Nicolas | author-link = Nicolas Bourbaki
 | contribution = 5. Characteristic exponent of a field. Perfect fields
 | contribution-url = https://books.google.com/books?id=GXT1CAAAQBAJ&pg=RA1-PA7
 | doi = 10.1007/978-3-642-61698-3
 | page = A.V.7
 | publisher = Springer
 | title = Algebra II, Chapters 4–7
 | year = 2003
| isbn = 978-3-540-00706-7 }}</ref>

Any field {{math|''F''}} has a unique minimal [[Field extension|subfield]], also called its [[prime field]]. This subfield is isomorphic to either the [[rational number]] field <math>\mathbb{Q}</math> or a finite field <math>\mathbb F_p</math> of prime order. Two prime fields of the same characteristic are isomorphic, and this isomorphism is unique. In other words, there is essentially a unique prime field in each characteristic.

=== Fields of characteristic zero ===
The fields of ''characteristic zero'' are those that have a subfield isomorphic to the field {{tmath|\Q}} of the [[rational number]]s. The most common of such fields are the subfields of the field {{tmath|\C}} of the [[complex number]]s; this includes the [[real number]]s <math>\mathbb{R}</math> and all [[algebraic number field]]s.

Other fields of characteristic zero are the [[p-adic field]]s that are widely used in number theory.

Fields of [[rational fraction]]s over the integers or a field of characteristic zero are other common examples.
 
[[Ordered field]]s always have characteristic zero; they include <math>\mathbb{Q}</math> and <math>\mathbb{R}.</math>

=== Fields of prime characteristic ===
The [[finite field]] {{math|GF(''p''{{sup|''n''}})}} has characteristic {{math|''p''}}.

There exist infinite fields of prime characteristic. For example, the field of all [[rational function]]s over <math>\mathbb{Z}/p\mathbb{Z}</math>, the [[algebraic closure]] of <math>\mathbb{Z}/p\mathbb{Z}</math> or the field of [[formal power series|formal Laurent series]] <math>\mathbb{Z}/p\mathbb{Z}((T))</math>. 

The size of any [[finite ring]] of prime characteristic {{math|''p''}} is a power of {{math|''p''}}. Since in that case it contains <math>\mathbb{Z}/p\mathbb{Z}</math> it is also a [[vector space]] over that field, and from [[linear algebra]] we know that the sizes of finite vector spaces over finite fields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power.{{efn|It is a vector space over a finite field, which we have shown to be of size {{math|''p''<sup>''n''</sup>}}, so its size is {{math|(''p''<sup>''n''</sup>)<sup>''m''</sup> {{=}} ''p''<sup>''nm''</sup>}}.}}