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If $F$ is a countable field, then proving that $F$ has algebraic closure is quite simple: there can be at most countable number of monic irreducible polynomials over $F$, let they form the set $\mathcal{L}$. For each polynomial in $\mathcal{L}$, we get an algebraic extension. Writing $\mathcal{L}$ as countable ascending union of finite subsets, one can obtain algebraic closure of $F$.

Of course the argument will not work if $\mathcal{L}$ is uncountable, some variation in argument is required. This already has been appeared in some other questions on mathstack.

But this raised one question to me:

Question: Does there exists a field $F$ with the following three properties:

(1) $F$ is uncountable

(2) $F$ is not algebraically closed.

(3) The number of monic irreducible polynomials of degree $>1$ is countable.

Note that $\mathbb{R}$ satisfies (1) and (2) but not (3).

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    $\begingroup$ For an irreducible polynomial $p(x)$, we can consider $p_{\lambda}(x)=\lambda^{-deg(p)}p(\lambda x)$. If $p(x)$ is irreducible, then I believe $p_{\lambda}(x)$ is also, and creates a $|F|$ number of polynomials. $\endgroup$
    – Pax
    Jun 22, 2016 at 2:29
  • $\begingroup$ I also expect it to be monic. is it monic? $\endgroup$
    – p Groups
    Jun 22, 2016 at 2:34
  • $\begingroup$ If you start with a monic $p(x)$, then $p_\lambda(x)$ will also be monic. $\endgroup$
    – Ted Shifrin
    Jun 22, 2016 at 2:37
  • $\begingroup$ perhaps you are right. $\endgroup$
    – p Groups
    Jun 22, 2016 at 2:41
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    $\begingroup$ Your argument for the countable case is incomplete. Writing $L$ as the increasing union of its finite subsets does not help you; you need to pick an order in which to adjoin the roots of irreducible polynomials. Once you do this there's no need for $L$ to be countable; just use the axiom of choice to pick a well-ordering of $L$. $\endgroup$
    – Qiaochu Yuan
    Jun 22, 2016 at 4:03

1 Answer 1

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There is no such field.

Take any uncountable field $F$ that is not algebraically closed.

Let $K$ be an algebraic closure of $F$.

Then $[K:F] > 1$ and hence $K \smallsetminus F$ is uncountable.

If there are countably many monic irreducible polynomials over $F$ of degree more than $1$, then there are countably many elements in $K \smallsetminus F$, since each of them has a minimal polynomial (which has finite degree), which gives a contradiction.

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