Alright, a little motivation for my question before I ask it... We have been assigned to read a section of our book (Anadijiban Das' "Tensors", section 1.1) and find the definitions of all words we don't know. The section I have been assigned is all about Fields. It gives the definition of a field and the axioms that constitute a field. Now, I understand, in a very basic way, what a field is. I can prove, using the axioms, whether or not some set constitutes a field. What information I am really looking for is a deeper understanding of a field and specifically how they relate to a vector space or tensors (and I do know that a vector space presupposes some field F).