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Difference between revisions of "Algebraic number field"

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An algebraic number field $K$ of degree $n$ is an extension of degree $n$ of the field $\mathbf Q$ of rational numbers. Alternatively, a number field $K$ is an algebraic number field (of degree $n$) if every $\alpha\in K$ is the root of a polynomial (of degree at most $n$) over $\mathbf Q$.  (Cf. also [[Algebraic number]]; [[Algebraic number theory]]; [[Extension of a field]]; [[Number field]].)
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Weiss,  "Algebraic number theory" , McGraw-Hill  (1963)  pp. Sects. 4–9</TD></TR>
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====Comments====
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Examples include:
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* [[Quadratic field]] — an extension of degree $n=2$;
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* [[Cyclotomic field]] — an extension generated by roots of unity.

Latest revision as of 19:55, 21 December 2015

2020 Mathematics Subject Classification: Primary: 11R04 [MSN][ZBL]

An algebraic number field $K$ of degree $n$ is an extension of degree $n$ of the field $\mathbf Q$ of rational numbers. Alternatively, a number field $K$ is an algebraic number field (of degree $n$) if every $\alpha\in K$ is the root of a polynomial (of degree at most $n$) over $\mathbf Q$. (Cf. also Algebraic number; Algebraic number theory; Extension of a field; Number field.)

References

[1] E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9

Comments

Examples include:

How to Cite This Entry:
Algebraic number field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_number_field&oldid=30303