[throwawaymath]

Huh? A scalar is very specifically a field element by definition. This is why it's important to specify the field you're working with when you talk about a vector space - a scalar is not going to be a real or a complex if your field isn't R or C.

If you've seen someone define a scalar as a real number, that's really only because they're informally stating their underlying field is R.

[ajross]

I keep seeing you people in this thread and wondering, with all respect, what planet you're coming from.

The whole purpose of this exercise is to see if there was a way to come up with a straightforward, reasonably informal, multiple choice question that would expose a fundamental understanding in basic university math concepts like "vector space" in the same way we see in primary math.

And instead all you people want to do is natter over the ways in which someone could cleverly make the "wrong" answer right. It's... beyond missing the point, it's actively working against the whole goal of the exercise.

[gizmo686]

Because you are going to have students who mark the answer as correct, and you need to be prepared to explain to them why it is wrong. In addition, you explanation of why it is wrong should be accurate, and should not suggest that other correct answers are also wrong. Returning to the original question, why is it that Z3 is not a vector space, but Q3 is. If you say that neither of these are vector spaces, then you have a misunderstanding about what a vector space is which the question would miss because the author forgot to include Q3 as an option.

By itself, this is a minor complaint (you cannot include every example in you choices, although I do think that an example which could not be viewed as an R-vector space would be good to include). However, when you explain why Z3 is not a vector space, your explanation must be correct. An explanation which also excludes Q3 is incorrect.