# vector spaces over modulus fields

I am in a linear algebra class and we are covering vector spaces over fields. We haven't been exposed to this material much about modulus and I cannot find it in my textbook and barely online. We are asked to determine whether the Integers modulus 8 over the field integers modulus 2 is a vector space or if the integers modulus 2 over the field integers modulus 8 is a vector space. I am confused on where to start here. In my head I feel like they both are, because I feel like the additive axioms all hold (zero vector in both is 0, and additive inverse in both is just whatever would sum to be 8 or 2, respectively). I feel like scalar multiplication fails somehow in one of these cases but I don't really know how. We were told as a hint that one of them isn't, but I can't figure out which. Any pointers?

• Welcome to Mathematics Stack Exchange. Integers modulo $8$ are not a field. A field cannot have non-zero zero divisors, but $2\times4=0$ in integers modulo $8$ Jan 27, 2021 at 0:55
• I see what you are saying... The integers modulo 8 cannot be a field because the elements are not relatively prime here? I am confused why this was phrased as a question though... We were asked "the integers modulo 2 over the integers modulo 8. " and determining the vector space over the given field. So this question is just not formulated right because it isn't even a field? Jan 27, 2021 at 0:57
• I still don't understand if the first is a vector space :( Jan 27, 2021 at 2:02
• In a $\mathbb{Z}_2$ vector space every vector satisfies $x+x=2\cdot x= 0\cdot x=0$. Does this hold in the abelian group $\mathbb{Z}_8$? Jan 27, 2021 at 7:49
• I would love to answer but honestly I feel like this class hasn't taught me anything. We only have had 2 lectures and our book only talks about vector spaces, nothing about abelian groups. Although I looked it up and I think it makes sense but I can't figure out what you mean by that. Are you talking about when the integers modulo 8 are a vector space over the field of integers modulo 2? I just don't think I understand the basics but can't find much online about it :( Our teaching hasn't taught it too well Jan 27, 2021 at 14:54

Integers modulus 2 are a field (the smaller field), so, one can speak about vector spaces over integer modulus 2. However, you need to check the distributivity of scalar multiplication and addition $$(t+s)\cdot v=t\cdot v + s\cdot v$$ Take $$t=s=1$$, integer modulus 2, and $$v=3$$, integer modulus 8. Note that $$t+s=0$$ modulus 2, so $$(t+s)\cdot v=0$$. But $$v+v=6$$ not zero modulus 8.