How is the trace pairing function $(x,y) \mapsto Tr(xy)$ on a number field an analogue of the dot product in euclidean space?
(This is a view shared by Keith Conrad and can be found in his notes Discriminants... and The Different Ideal)
How is the trace pairing function $(x,y) \mapsto Tr(xy)$ on a number field an analogue of the dot product in euclidean space?
(This is a view shared by Keith Conrad and can be found in his notes Discriminants... and The Different Ideal)
The vector space of $n$ by $n$ matrices with real entries has a positive definite inner product given in the convenient shape $ \langle A, B \rangle = \; \mbox{tr} \; \left(A B^T \right) $ which is pretty much what I am seeing in his notes.
Multiplication in a field is linear over the base field, so you get an analogy. Depending how you want to order things, let the $e_i$ be a basis for your field over the base field. Then multiplication by some field element $x$ is completely determined by the matrix of values $x_{ij}$ such that $ x e_i = \sum x_{ij} e_j. $ So there is your matrix.
I've got to say, for utter improvisation, this is pretty good.