An algebraic structure, the most familiar instance of which is the real numbers. A field is a set F with two binary operations (functions from pairs in F into F), marked + and ., which we call addition and multiplication -- in the case of reals, these are the familiar operations. The . is often dropped. The following properties hold:
- There exist (different) elements 0 and 1, such that for all a in F a.1 = a+0 = a
- Associativity: for all a,b,c in F a+(b+c) = (a+b)+c and a(bc)=(ab)c
- Commutativity: for all a,b a+b = b+a and ab = ba
- Distributivity: for all a,b,c in F a(b+c) = (ab) + (ac)
- Inverse elements: for every a in F there exists an element marked -a such that -a+a=0; if a is not zero, there is also an element marked a-1 such that aa-1=1
The complex numbers are another important and well-known field. So are finite fields.
Galois Theory connects fields and their subfields (or extension fields) with their automorphism groups and their subgroups. It is the source of much of what we can say about fields...