Fields and their relation to Tensors

[iamalexalright]

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).

 

[Hurkyl]

Hrm. My first inclination is to associate them in the exact opposite way. Vectors and stuff tell you about the arithmetic of your field, rather than the field telling you stuff about vectors.

 

[iamalexalright]

Alright, I can somewhat see what you are saying (keep in my mind this is really my first time being exposed to these subjects). Could you perhaps expound?