Is QQ^n a vector space over ZZ or over QQ?

Jaxson Mack

Jaxson Mack

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2022-08-18

Is Q n a vector space over Z or over Q ?
Q n is clearly not a vector space over R , because scalar multiplication of some q Q by π renders π q, which is irrational.

Answer & Explanation

Joe Sheppard

Joe Sheppard

Beginner2022-08-19Added 12 answers

You can't have a vector space over Z . By definition, a vector space is required to be over a field. If you take away the field requirement, what you're left with is calles a module. And yes, Q n is definitely a Z -module.
That being said, there is a list of 10 requirements (or thereabouts, it varies slightly) for a space like Q n to be a vector space over a field like Q . All of them should be easily verifiable. So yes, Q n is a vector space over Q .

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