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Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$?

That is, if a field is of characteristic 2, then does this field have to be $\{0,1\}$?

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  • $\begingroup$ You may not have expected so while asking the question, but as a matter of fact the tag (finite-fields) does not fully apply as there are infinite fields of charateristic $2$ (as per Asaf's and Chris's answers) $\endgroup$ – Hagen von Eitzen Jul 7 '13 at 13:59
  • $\begingroup$ @HagenvonEitzen: It was in fact Asaf who applied that tag. $\endgroup$ – Chris Eagle Jul 7 '13 at 14:05
  • $\begingroup$ Take a look at my answer to "Example of infinite field of characteristic $p\neq 0$". $\endgroup$ – Zev Chonoles Jul 7 '13 at 15:02
  • $\begingroup$ @ChrisEagle Tsk, tsk, shame on him. :) $\endgroup$ – Hagen von Eitzen Jul 7 '13 at 15:35
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It is not hard to see that $x^2+x+1$ does not have a root in $\Bbb{Z/2Z}$. Therefore we can extend the field with this root. Also every field has an algebraic closure, and this shows that $\Bbb{Z/2Z}$ is not algebraically closed, so its closure is strictly larger (and in fact infinite).

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    $\begingroup$ (And the characteristic of any extension of a field is the same as that of the base field since they share the same multiplicative identity element.) $\endgroup$ – user714630 Jul 7 '13 at 12:54
  • $\begingroup$ Anyway, there are infinitely many different fields of characteristic 2, but they all embed $\mathbb Z/2 \mathbb Z$. $\endgroup$ – Johannes Kloos Jul 7 '13 at 13:17
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The field of rational functions with coefficients in $\Bbb{Z}/2\Bbb{Z}$ is an easy example of an infinite field of characteristic $2$.

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To a beginner, knowing how one could think of an answer is at least as important as knowing an answer.

For examples in Algebra, one needs (at least) two things: A catalogue of the basic structures that appear commonly in important mathematics, and methods of constructing new structures from old. Your catalogue and constructions will naturally expand as you study more, so you don't need to worry about this consciously. The moral I am trying to impart is the following: Instead of trying to construct a particular a structure with particular properties "from scratch" (like I constantly tried to when I was starting to learn these things), first search your basic catalogue. If that doesn't turn up anything, more often than not your search will hint at some basic construction from one of these examples that will work.

When you start learning field theory, your basic catalogue should include all of the finite fields, the rationals, real numbers, complex numbers, and algebraic numbers. Your basic constructions should be subfields, field extensions, fields of fractions and algebraic closures. You should also have the tools from your basic ring theory; constructions like a quotient ring and ring extensions also help with this stuff.

For example, Chris came up with his answer by starting with the easy example of a field of characteristic two and wanting to make it bigger. So he extended it with an indeterminate $X$ and as a result he got the field of rational functions with coefficients in $\mathbb{Z}/(2).$ Asaf suggested two ways: making it bigger by taking the algebraic closure, or extending the field by a root of a polynomial (I personally like to see this construction as a certain quotient ring).

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Just to expand Asaf's answer, take polynomials in $x$, modulo $x^2+x+1$. The coefficients are in $\mathbb{Z}/2\mathbb{Z}$. They look like linear polynomials in $x$. That gives a field of four elements. If you can find a cubic $C(x)$ which is nonzero $(\mod 2)$ for $x=0$ and $x=1$, you can take polynomials modulo $C(x)$, which is all the quadratic polynomials, and is a field of eight elements.

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