Discriminant of a number field

Annoying index

Given a number field 𝐾 with ring of integers O𝐾, and a ℚ-basis {𝛼𝑖}𝑛𝑖=1 ⊂O𝐾 of 𝐾, we call the quantity

∣O𝐾ℤ𝛼1+⋯+ℤ𝛼𝑛∣

the annoying index,¹ alg since it measures the degree to which {𝛼𝑖}𝑛𝑖=1 fails to be an Integral basis for 𝐾

Properties

1. If Δ𝐾:ℚ(𝛼1,…,𝛼𝑛) is squarefree, then {𝛼𝑖}𝑛𝑖=1 is an integral basis.
2. If the minimal polynomial 𝑚𝛼(𝑥) for 𝛼 ∈O𝐾 is Eisenstein at 𝑝, then 𝑝 does not divide |O𝐾/ℤ[𝛼]|.

Proof

^P1 follows immediately from ^EQ1.

For ^P2, suppose towards contradiction that 𝑝 divides the annoying index. Then by Cauchy’s order theorem there exists 𝜉 ∈O𝐾 such that 𝜉 +ℤ[𝛼] has order 𝑝, i.e. 𝜉 ∉ℤ[𝛼] and 𝑝𝜉 ∈ℤ[𝛼]. It follows we can write

𝑝𝜉=𝑛−1∑𝑖=0𝑏𝑖𝛼𝑖

for some 𝑏𝑖 ∈ℤ where some 𝑏𝑗 is not divisible by 𝑝. Fix 𝑗 to be the smallest such index. It follows

𝜂=𝜉−𝑗−1∑𝑖=0𝑏𝑖𝑝𝛼𝑖=𝑛−1∑𝑖=𝑗𝑏𝑖𝑝𝛼𝑖∈O𝐾.

and so

𝜂𝛼𝑛−𝑗−1=𝑏𝑗𝑝𝛼𝑛−1+𝛼𝑛𝑝𝑛−𝑗−2∑𝑖=0𝑏𝑖+𝑗+1𝛼𝑖∈O𝐾.

Since

𝛼𝑛𝑝=−𝑛−1∑𝑖=0𝑎𝑖𝑝𝛼𝑖∈O𝐾

it follows 𝛽 =𝑏𝑗𝑝𝛼𝑛−1 ∈O𝐾, so the field norm N𝐾:ℚ⁡(𝛽) ∈ℤ.

On the other hand

N𝐾:ℚ⁡(𝛽)=𝑏𝑛𝑗𝑝𝑛N𝐾:ℚ⁡(𝛼)𝑛−1=±𝑏𝑛𝑗𝑎𝑛−10𝑝𝑛∉ℤ

since 𝑝 ∤𝑏𝑗 and 𝑝2 ∤𝑎0, a contradiction.

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Footnotes

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