Let $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$ and $\bar{\mathbb{Q}_{\ell}}$ the algebraic closure of $\mathbb{Q}_{\ell}$ ($\ell$ is an integer). Could we describe the elements in $\bar{\mathbb{Q}}$ and $\bar{\mathbb{Q}_{\ell}}$ explicitly? Here $\mathbb{Q}_{\ell}$ is the $p$-adic field at $\ell$. Thank you very much.
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$\overline{\Bbb Q}$ is the set of algebraic numbers: complex numbers that are solutions to polynomials with rational coefficients. That's about as explicit as I can give. Would a similar explanation of $\overline{\Bbb Q}_\ell$ suffice? – Mar 19 '14 at 13:10
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@Mike, thank you very much. Is $\bar{\mathbb{Q}}$ strictly contained in $\mathbb{C}$? – LJR Mar 19 '14 at 13:16
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Yes, since $\Bbb Q \subset \Bbb C$ and $\Bbb C$ is algebraically closed. (Of course, the algebraic closure is an entirely algebraic object, but the algebraic closure of $\Bbb Q$ (abstractly) is isomorphic to the field of algebraic numbers (a subset of $\Bbb C$). This is probably not an issue that matters to you.) – Mar 19 '14 at 13:28
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@Mike, thank you very much. – LJR Mar 19 '14 at 13:31
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The fields $\overline{\mathbb{Q}_{\ell}}$ and $\mathbb{C}$ are indeed isomorphic as abstract fields, i.e., both are algebraically closed fields of characteristic zero and with transcendence degree $2^{ℵ_0}$ over $\mathbb{Q}$. Of course, this isomorphism does not preserve topology: $\overline{\mathbb{Q}_{\ell}}$ is not complete, for example, but $\mathbb{C}$ is. The completion of of $\overline{\mathbb{Q}_{\ell}}$ is $\mathbb{C}_{\ell}$. There are many desriptions about the algebraic closures, but it depends on what you want to know. Perhaps you find also the following question interesting in this context: How far are the $p$-adic numbers from being algebraically closed?.
Dietrich Burde
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