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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?

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  • $\begingroup$ 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 $8$ $\endgroup$ Jan 27, 2021 at 0:55
  • $\begingroup$ 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? $\endgroup$
    – DM Zach
    Jan 27, 2021 at 0:57
  • $\begingroup$ I still don't understand if the first is a vector space :( $\endgroup$
    – DM Zach
    Jan 27, 2021 at 2:02
  • $\begingroup$ 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 $\mathbb{Z}_8$? $\endgroup$ Jan 27, 2021 at 7:49
  • $\begingroup$ 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 $\endgroup$
    – DM Zach
    Jan 27, 2021 at 14:54

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A vector space has two operations: a internal one (addition), and a external one which, by definition, requires a field (like the fields of real or complex numbers, any finite field …)

Integers modulus 8 are not a field, so, you can not speak about vector spaces over integers modulus 8 (there is another name for some analogous struture but it is not “vector space”). When you work modulus 8, there is no way to “invert”, for instance, 4, or 6. To get a field you must work with integers modulo a prime number … For instance, modulus 5 you can invert 4, since 4x4=16=1mod5

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.

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