Discriminant of a number field

Annoying index

Given a number field with ring of integers , and a -basis of , we call the quantity

the annoying index,¹ alg since it measures the degree to which fails to be an Integral basis for

Properties

1. If is squarefree, then is an integral basis.
2. If the minimal polynomial for is Eisenstein at , then does not divide .

Proof

^P1 follows immediately from ^EQ1.

For ^P2, suppose towards contradiction that divides the annoying index. Then by Cauchy’s order theorem there exists such that has order , i.e. and . It follows we can write

for some where some is not divisible by . Fix to be the smallest such index. It follows

and so

Since

it follows , so the field norm .

On the other hand

since and , a contradiction.

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Footnotes

1. This term is taken from lectures by Florian Breuer and should not be taken too seriously. ↩