Number field

Discriminant of a number field

Let be a number field of degree and be an Integral basis. The discriminant of is given by alg

where the latter quantity is the discriminant of a separable extension and is an integer independent of the choice of integral basis.¹

Proof

Suppose and are both integral bases for . Let and . We can find an appropriate change of basis matrix such that

whence

Now since and are both integers, it follows . Thus , as required.

Since is a product of algebraic integers, it follows .

For a general -basis , we have

where all operands are integers. We call the index on the right had side the Annoying index.

Proof

Suppose are an integral basis of and write

for some . Let . By Subgroup of a free abelian group, . The fact that

yields the desired result.

See also Discriminant of an algebraic integer.

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Footnotes

1. 2022. Algebraic number theory course notes, ¶¶2.2–2.3, p. 34 ↩