Examples of fields with characteristic $2$. [closed]

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What are good examples of fields of characteristic $2$, starting from the simplest one to more interesting examples?

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2 Answers

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The simplest is $\mathbb{F}_2$, the field with two elements. The next simplest are its finite extensions, the finite fields $\mathbb{F}_{2^n}$ of characteristic $2$, which together make up its algebraic closure $\overline{\mathbb{F}_2}$. The algebraic closure has various other subfields beyond the finite ones, which can be classified using Galois theory.

Next, there's $\mathbb{F}_2(t)$. Its finite extensions include the function fields over $\mathbb{F}_2$, which describe algebraic curves over $\mathbb{F}_2$, as well as function fields over $\mathbb{F}_{2^n}$. I'm less familiar with the algebraic closure of this thing but probably there are nice things to say about it.

And so forth. In general, every field of characteristic $2$ is an algebraic extension of a purely transcendental extension of $\mathbb{F}_2$, although this is less helpful than it sounds.

A very interesting example, which is so large that it does not form a set, is the "field" of nimbers.

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The simplest field with characteristic $2$ is $F_2 = Z/2Z$.

More generally, for every $n \geq 1$, there is, up to isomorphism, exactly one field of cardinality $2^n$, denoted $F_{2^n}$. Like all fields of characteristic $2$, these fields contain $F_2$ as a subfield.

Given any field $K$ of characteristic $2$, the algebraic closure $\tilde{K}$ of $K$ also has characteristic $2$. For example, the algebraic closure of $F_2$ can be considered the union of all the fields $F_{2^n}$.

Given any field $K$ of characteristic $2$, the field of rational functions $K(X)$ with coefficients in $K$ also has characteristic $2$. So does the field $K((X))$ of formal Laurent series over $K$.

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