I am trying to understand about completeness and order of fields. Does it make sense to talk about completeness of finite fields? I know finite fields are not ordered in a way compatible with addition and multiplication. But are they complete? A finite field of three elements, won't satisfy the equation $x^2+1=0$. So I think it should be incomplete. But how are Cauchy sequences in finite fields defined?
Completeness of finite field
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finite-fields
cauchy-sequences
complete-spaces
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3Also note that a metric on a finite set can only produce the discrete topology anyway, and in a discrete space, a sequence $(a_n)$ converges to $a$ if and only if eventually all terms are equal to $a$. – 2017-02-17
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0So, does this mean that by just the existence of a ordered field, one cannot talk about its completeness unless one defines a metric on that field? And is real number line complete ordered field only on usual metric?? – 2017-02-17
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2The only topologies on a finite field that make the field operations continuous are the discrete topology and the trivial topology. The latter is not Hausdorff, so convergence of sequences is not really a thing. With respect to the discrete topology (or the discrete metric) the finite field is complete, because all Cauchy sequences are eventually constant (and therefore not very interesting). – 2017-02-17
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1I don't understand why you bring up solvability of $x^2+1=0$. It has no solutions in $\Bbb{R}$ either, yet $\Bbb{R}$ is complete w.r.t. the absolute value metric. Algebraic and topological closure are not totally unrelated (odd degree polynomials have a zero in $\Bbb{R}$, look up real closed fields), but they are separate and very different concepts. – 2017-02-17
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0@jnyan: It seems to me that you are mixing the different concepts of "Cauchy completeness" (which is about nets and their limits) and "algebraic completeness" (which is about roots of algebraic equations). You should clarify what you mean if you want to receive an answer. – 2018-04-17