Difference between a vector space and a field

[torquerotates]

One book defined a vector space as a set of objects that can undergo the laws of algebra "over" the field of scalars. But doesn't the laws of algebra also hold in a field? If so, wouldn't a field be a vector space also? Wouldn't that make the definition of a vector space meaningless as it uses circular logic?

 

[morphism]

That's a good observation. Every field is in fact a vector space over itself or over any of its subfields. But of course the converse isn't true, i.e. there are vector spaces which aren't fields. The reason being that while in a field you must be able to "multiply" the elements together (and not just multiply by scalars), there is no such requirement in a vector space.

 

[n_bourbaki]

One book defined a vector space as a set of objects that can undergo the laws of algebra "over" the field of scalars. But doesn't the laws of algebra also hold in a field?

Presumably, but it is slightly ambiguous phrase.

If so, wouldn't a field be a vector space also? Wouldn't that make the definition of a vector space meaningless as it uses circular logic?

No, there is nothing circular, and it certainly isn't circular logic. At worst it would be 'redundant', but since vector spaces are not fields it isn't.

 

[HallsofIvy]

Given a field, F, you certainly can define F to be a vector space over itself. That's not often done because you don't learn anything new. The dimension would, of course, be 1.

However, since the definition of "vector space" allows for the vectors to be different from the underlying field, no, the definition is not circular.