Difference between revisions of "Algebraic number field"
From Encyclopedia of Mathematics
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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 theory]]; [[ | + | |
| + | 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==== | ====References==== | ||
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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> | <TR><TD valign="top">[1]</TD> <TD valign="top"> E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9</TD></TR> | ||
</table> | </table> | ||
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| + | ====Comments==== | ||
| + | Examples include: | ||
| + | * [[Quadratic field]] — an extension of degree $n=2$; | ||
| + | * [[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:
- Quadratic field — an extension of degree $n=2$;
- Cyclotomic field — an extension generated by roots of unity.
How to Cite This Entry:
Algebraic number field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_number_field&oldid=35749
Algebraic number field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_number_field&oldid=35749