Algebraic closure

English

Noun

1. Template:Lb A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G).
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Usage notes

• Notations for the algebraic closure of a field $F$ include $\bar{F}$ and $F^{\mathrm{a}}$.
• Using Zorn's lemma (or the weaker Template:W), it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Consequently, authors often speak of the (rather than an) algebraic closure of K. (See Template:Pedia)
• The field of complex numbers, $ℂ$, is the algebraic closure of the field of real numbers, $ℝ$.
• The algebraic closure of the field of p-adic numbers, ${\mathbb{Q}}_p$, is denoted ${\bar{\mathbb{Q}}}_p$ or ${\mathbb{Q}}_p^{\mathrm{a}}$. (Unlike $ℂ$, and indeed unlike ${\mathbb{Q}}_p$, ${\bar{\mathbb{Q}}}_p$ is not metrically complete: its metric completion, which is algebraically closed, is denoted ${ℂ}_p$ or ${\mathrm{\Omega }}_p$.)

Related terms

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Translations

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References

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Further reading

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• Algebraic closure on Template:W
• Algebraic Closure on Template:W