Last updates: 29 May 2011
Note that every field is a commutative ring and the only conditions in the defnition of a field that are not in the definition of a ring are (f) and (h).
Important examples of fields are:
Field homomorphisms might be used to compare fields. The only problem is that there aren't many interesting field homomorphisms, as we show in Proposition (fldhominj). We shall study fields in more depth in Part V.
Let and be fields with identities and , respectively.
HW: Show that if is a
field homomorphism then , where
are the zeros in and
HW: Explain why conditions (a) and (b) in the definition of a field homomorphism do not imply condition (c).
If is a field homomorphism then is injective.
Proposition (fldhominj) stated another way, says that the kernel of any field homomorphism is . This means that we cannot get an interesting analogue of Theorem ??? for fields. Proposition (fldhominj) also shows that if is a field homomorphism then is a subfield of .
These notes are written to highlight the analogy between groups and group actions, rings and modules, and fields and vector spaces.
[Ram] A. Ram, Notes in abstract algebra, University of Wisconsin, Madison 1993-1994.