Discriminant of a number field

Discriminant of an algebraic integer

Let be an algebraic integer of degree with minimal polynomial and . The discriminant of is then alg

where is the field norm and is the formal derivative.

Proof

Let , be the distinct embeddings of in , and thus be the conjugates of . Expanding the definition of the field norm,

where since

it follows

and thus

Now the term being squared is precisely the determinant of the Vandermonde matrix

therefore .

In particular, if the minimal polynomial is of the form

then we have

Proof

Let . Since annihilates , we have

whence and . Now since

we can multiply by to get

whence

is a monic annihilating polynomial for , which must be minimal since . Again invoking the fact , we have

whence .

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