HOMEWORK 1
DUE: 2/12
Please do the following exercises from Marcus: Chapter 2: 8, 22, 28, 29, 40, 41.
Problem A. (basically Exercise 27) Let K/Q be a number field extension.
(a) Given an inclusion M ⊂ N of Z-modules, if N/M is a finite Z-module, then recall that it’s order
[N : M ] := |N/M | is the index of M in N . Show that M ⊂ OK is an algebraic lattice in K if and
only if M is of finite index in OK .
(b) Note that the discriminant disc(M ) of any algebraic lattice M ⊂ K (by which I mean the discriminant
of any integral basis) is an invariant of M ⊂ K (not of M abstractly!). Show that for any inclusion
M ⊂ N of algebraic lattices in K,
disc(M ) = [N : M ]2 disc(N )
(c) Suppose R ⊂ K is a subring. Show that R is finitely generated as a Z-module if and only if R ⊂ OK .
(d) An order of the number field K is a subring R ⊂ K that is an algebraic lattice. Show that R ⊂ OK
and
disc(R) = [OK : R]2 disc(OK )
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Some notes on nil-semicommutative rings