Vector spaces - "over a field" I recently learned about vector spaces, and had two questions:

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*What do we mean by saying “vector space over a field”? I read many posts and listened to lectures but it seems to me that people have different definitions/explanations for the terminology “over a field” – in one lecture, the professor said that when we say a vector space "over a field" we mean that the scalars with which we multiply the elements of the vector space are taken from some field F. In another lecture, a professor said that “over a field” means that the components of the elements in the vector space are from some field F.
After hearing their explanations, I got confused – does the terminology “over a field” refers to the elements of the vector space themselves or to the scalars we multiply the vectors in the V.S. with?

I would appreciate it if you could explain the terminology of “over a field” in more detail than those professors did.


*When talking about a polynomial over a field F, we mean that the coefficients of the polynomial are taken from some field F. My question is the following: do the roots of the polynomial are from the field F because the coefficients belong there too, or in other words – does the choice of a field of the coefficients determines the field to which the roots of the polynomial belong?

Sorry about the bad English (I'm not a native speaker) and the silly question, I just started learning linear algebra so it’s still new and hard for me…
Thank you very much for reading my question!
 A: This circles back to the definition of a vector space:
a vector space $V$ over a field $F$ is a set with two operations $+ : V \times V \to V$ and $\cdot : F \times V \to V$ such that

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*$(V,+)$ is an abelian group

*$+, \cdot$ are compatible with the field operations: $1 \cdot v = v$, $(ab) \cdot v = a \cdot (b \cdot v)$, $a \cdot (u + v) = a \cdot v + a \cdot v$, $(a + b) \cdot v = a \cdot v + b \cdot v$
So this "over some field" is baked right into the definition - it just means "the elements we can multiply our vectors by". For example if you take the vector space $\mathbb{R}^2$ with the usual operations you can multiply each element by real numbers and thus it's a vector space over $\mathbb{R}$, but you could also multiply them by rational numbers which means it's also a vector space over $\mathbb{Q}$.
A: *

*When we say that $V$ is a vector space over a field $F$, what that means is your first option: that the scalars are taken from $F$. For instance $\Bbb R$ is a $1$-dimensional vector space over $\Bbb R$, and an infinite-dimensional vector space over $\Bbb Q$.

*It needs context. If we say that $P(x)$ has no roots, it is a good idea to add where it has no roots. For instance $x^2+1\in\Bbb R[x]$ and it has no roots in $\Bbb R$, but it has roots in $\Bbb C$.

