I just started my first upper level undergrad course, and as we were being taught vector spaces over fields we quickly went over fields. What confused me that the the set {0, 1, 2} was a field. However, to my understanding, that set doesn't satisfy the axiom, "For every element a in F, there is an element b such that a+b=0", among others. Can someone help clarify where my understanding is off.Also one my friend states "for every prime power p^n, there exists a field with p n elements" and then doesn't expand on it. If someone could give an example or proof i would be very grateful.

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I just started my first upper level undergrad course, and as we were being taught vector spaces over fields we quickly went over fields. What confused me that the the set {0, 1, 2} was a field. However, to my understanding, that set doesn&#039;t satisfy the axiom, &quot;For every element a in F, there is an element b such that a+b=0&quot;, among others. Can someone help clarify where my understanding is off.Also one my friend states &quot;for every prime power p^n, there exists a field with       p    n   elements&quot; and then doesn&#039;t expand on it. If someone could give an example or proof i would be very grateful.

bojnimaspnek · Accepted Answer

{0,1,2}  is a field if you do all arithmetic modulo 3, that is, adding/subtracting an appropriate multiple of 3 after each operation to make the result one of 0, 1 or 2.In aritmetic modulo 3, the negative of 1 is 2, because 1+2=3 and subtracting 3 to get that into the range {0,1,2} makes 0.This construction will give you the fields with p elements. Getting to the fields with       p    n   elements for   n  ≥  2 requires more algebraic prerequisites than will probably fit comfortably into a MSE answer.

I just started my first upper level undergrad course, and as we were being taught vector spaces over

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