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Exercise 3 from Roman's book "Advanced Linear Algebra".

The author asks us to "find an abelian group $V$ and a field $\mathbb{F}$ for which $V$ is a vector space over $\mathbb{F}$ in at least two different ways, that is, there are two different definitions of scalar multiplications making $V$ a vector space over $\mathbb{F}$."

I would appreciate any hint in order to solve this question.

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up vote 4 down vote accepted

Here's a hint: Pick any field $F$ with a nontrivial automorphism. That this, there is a nonidentity function $f:F\rightarrow F$ which is bijective, and such that $f(x+y) = f(x) + f(y)$ and $f(xy) = f(x)f(y)$ for any $x$ and $y$.

Let $V$ be any vector space over $F$. Then you can define a new vector space structure on $V$ by saying $\lambda \cdot v = f(\lambda)v$ where multiplication on the right refers to the original vector space structure.

I'll leave it to you to both find an example of such a field and show that this actually works.

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