Math 627
Homework #2
September 24, 2014
Due Friday, October 3
1. Let {x1 , x2 , . . . , xn } be independent variables over a field K. Prove that


1 x1 x21 · · · x1n−1
1 x2 x2 · · · xn−1 
Y
2
2


det  .. ..
(xj − xi ).
..
..  =
. .
.
.  1≤i<j≤n
1 xn x2n · · · xnn−1
Hint: look at both sides of the equations as polynomials of degree precisely n − 1
in one of the variables, say xn ; observe what the roots are, and compare the leading
coefficients.
This formula is called the Vandermonde determinant formula. Yes, you can probably
find dozens of proofs of it online—however, it’s worth working it out, and I recommend
you try it on your own first!
2. Let K/Q be an extension of degree n. Let α1 , . . . , αn ∈ OK be chosen so that they
also form a basis for K as a Q-vector space. Let
∆ = DQK (α1 , . . . , αn ).
(a) Show that ∆ is an integer multiple of the absolute discriminant DK of K.
(b) Show that {α1 , . . . , αn } is an integral basis for OK if and only if |∆| = |DK |.
3. Let K = Q(α) be an extension of degree n, and assume that α ∈ OK . Let D =
DQK (1, α, . . . , αn−1 ). For each i, 1 ≤ i ≤ n, define the subset of OK ,
1
i
(c0 + c1 α + · · · + ci α ) : cj ∈ Z, ci 6= 0 ,
Xi =
D
and let
1
i
mi = min |ci | : (c0 + c1 α + · · · + ci α ) ∈ Xi .
D
Then let
Mi =
1
i
β ∈ Xi : β = (c0 + c1 α + · · · + mi α ) .
D
For each i, now fix an element βi ∈ Mi . Show that {β1 , . . . , βn } is an integral basis
of OK .
1