Algebraic number field

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.