# Algebraic number field

From Encyclopedia of Mathematics

2010 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://www.encyclopediaofmath.org/index.php?title=Algebraic_number_field&oldid=37054