Modern Algebra Question

[Antrax]

From my Modern Algebra final:

4)
e) Prove/disprove: A multiplication can be defined on Z6 such that it becomes an integral domain.

It was a "new" question (meaning nothing similar was ever used on a test), so I'm not sure if I got it right. I showed it to Lili and she didn't solve it in two seconds, so I'm assuming it's of SOME interest.
Antrax
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[Antrax]

Here's my answer, invisibled:

Incorrect. Proof: If Z6 becomes an integral domain, it has a "1". Consider the expression ("1"+"1")*("1"+"1"+"1"). By the distributive rule we get "1"*"1"+"1"*"1"+"1"*"1"+"1"*"1"+"1"*"1"+"1"*"1"="1"+"1"+"1"+"1"+"1"+"1".
Here is the first possibly weak step:
This is equivalent to a^6 in the regular (Z6,+), and since 6 is the order of Z6, it means 1+1+1+1+1+1=0.
So we found a multiplication of two things from the ring that gives off zero. Now we'll prove neither of them is zero.
Let's suppose "1"+"1"=0. Then we get 2+2=2*"1"+2*"1"=0. But 2+2=4, contradiction. Similarily 1+1+1 can't be zero, QED.
I strongly doubt this is correct. Where are my errors, and what is the right proof?
Antrax
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After years of disappointment with get rich quick schemes, I know I'm gonna get rich with this scheme. And quick!

[kevinatilusa]

Looks fine, but your weak step can be shortened considerably:
If we made Z_6 into an integral domain, it would still have the additive structure of Z_6, so 1+1=2, not 0.

[Antrax]

kevinatilusa, I was walking on eggshells, having never seen something like this. I really didn't want to make a leap that wasn't mathematically valid, so I just wrote out everything.
Here's another question from the same test, the other non-trivial question:
4)
d) Can there be an integral domain of size 3, 4, 5? Prove.

Antrax
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After years of disappointment with get rich quick schemes, I know I'm gonna get rich with this scheme. And quick!

[Antrax]

Actually, I think the shortening isn't valid. Nobody said "1" is the 1 in Z6. The multiplication could be defined such that 3="1" (for example, a*b=1/3*a*b. Obviously this wouldn't work, but just as an example).
Antrax
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After years of disappointment with get rich quick schemes, I know I'm gonna get rich with this scheme. And quick!

[kevinatilusa]

Ah, I was reading your argument as taking "1" to be the "1" in the additive group instead of the multiplicative one. If we let `1` be the generator for the additive group, the argument goes through I think, since 3*2=(`1`+`1`+`1`)(`1`+`1`)=`1`*`1`+`1`*`1`+`1`*`1`+`1`*`1`+`1`*`1`+`1`*`1`=0 regardless of whether `1`*`1`=0 or not, since anything added to itself 6 times gives 0.

For the other question,
The integers mod 3 and the integers mod 5 give you two of the domains. For the other one, we have to work a little bit harder.

Suppose there is such a domain. What are the two groups? The additive group has order 4, so is either Z_4 or Z_2xZ_2. The former is impossible, since we would (by the same argument) have 2*2=(`1`+`1`)(`1`+`1`)=0. In the latter case, let's call the elements 0,1,a,b. The additive table looks like:

+ 0 1 a b
0 0 1 a b
1 1 0 b a
a a b 0 1
b b a 1 0

For the multiplicative table, we also have little choice, since the 0 and 1 rows and columns are decided, and there's only one way to fill in the remainder to avoid repeats in the columns

* 0 1 a b
0 0 0 0 0
1 0 1 a b
a 0 a b 1
b 0 b 1 a

A good bit of checking shows this works.

Random comments and technical stuff about the above:

Another way of constructing the size 4 domain is to take the polynomials with integer coefficients mod 2, and then mod out by x^2+x+1 (whenever we have a quadratic term or higher, we replace x^2 by -(x+1) which equals x+1 since we're working mod 2). Therefore (for example) (x+1)(x+1)=x^2+2x+1=x^2+1=(x+1)+1=x.

I picked x^2+x+1 for two reasons: First to get the right number of elements (2 possibilities for the constant term, 2 for the x coefficient, and we cancel out the rest). Secondly, x^2+x+2 doesn't factor mod 2. What this means is that if (x^2+x+2) divides a product of two things (i.e. if the product of those two things is 0 once we mod out) then it had to divide one of the original things (i.e one of our original things was 0 after modding out). This irreducibility guaranteed that we got an integral domain.

The same technique works for any power of a prime. If I want to construct an integral domain of size p^k, what I do is take the polynomials mod p, then mod out by some kth degree polynomial which doesn't factor. What I'm left with is the polynomials with coefficients between 0 and p-1, and at most k coefficients, of which there are exactly p^k. As long as I can find a polynomial of degree k which does not reduce mod p, this construction will give an integral domain. For example, x^3+x+1 doesn't factor mod 5 (if it did, one of the factors would be linear and would correspond to a root of the polynomial, which it doesn't have). Therefore if we take the polynomials mod 5 and mod out by x^3+x+1, we get an integral domain with 125 elements.

[Antrax]

Whoops. I used the multiplication table (with elements 0,1,a,b) to prove that it can't be done
Antrax
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After years of disappointment with get rich quick schemes, I know I'm gonna get rich with this scheme. And quick!