Bases for a number field

Let be a number field of degree . As an -dimensional -vector space, one may choose a basis for .

Types

Integral basis

An integral basis a -basis for (which always exists), #m/def/num/alg whence it is also a -basis for .

Power basis

A power basis is a basis of the form for some , #m/def/num/alg whose existence is guaranteed by the primitive element theorem.

Integral power basis

An integral basis which is also a power basis is called an integral power basis. #m/def/num/alg These need not exist: A number field possessing an integral power basis is called a Monogenic field.

General properties¹

Suppose is a (non-integral) -basis for , and let be the corresponding discriminant. Then span a -module containing .
If are a -bases for such that the discriminant is squarefree, then they form an Integral basis.

Proof of 1–2

Let . Then for and we just need to show that for all . Letting range over embeddings of we see

with defined as in Discriminant of a separable extension. Multiplying each side by and letting we have

for some , whence

so since is an algebraic integer for each . Since , it follows for each , proving ^P1 .

^P2 follows from the fact that the discriminant of algebraic integers is in an integer and ^EQ1.

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2022. Algebraic number theory course notes, §2.1 ↩