189-571B: Higher Algebra II
To be presented: Wednesday, January 24.
The problem marked with a (*) is for everyone to do and to hand in on
the day of the problem session.
I will return this problem to you on Friday of the same week,
so late problems will not be accepted!
If you are stuck on your problem, come see me during my Monday
office hours (or make an appointment)
to get a hint...
Let K be a field extension of a field F
of finite degree (n, say) over F.
(Note: in our terminology a
field is always commutative.)
a) Show that if K is a seperable quadratic extension of
F (so that n=2) then
the tensor product over F of K with itself is isomorphic to the
direct sum of K with itself -- as F-algebras, and also as K-algebras.
b) If K is seperable cubic extnesion of F, show that
the tensor product of K with K, over F, is isomorphic (both as F- or K-algebras)
K + K +K if K/F is Galois.
K + K', where K' is the normal closure of K over F, if K is not Galois over F.
2. [ Kumi Cardinal ]
(Structure of modules over non-commutative division rings.)
Let D be a (not necessarily commutative) division ring.
Show that every finitely generated module M over D is free, i.e., has a
basis. Use this to define the notion of dimension of M of D and prove the
dimension formula, for all sub-division rings D' of D:
dimD'(M) = dimD(M) dimD'(D)
generalising what was proved in class last semester for fields.
3. [ Kristina Loeschner ]
Show that the tensor product of H with itself (over R) is
isomorphic to M4(R) as an R-algebra.