Wikipedia:Reference desk/Archives/Mathematics/2019 January 15

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January 15

Algebraic integers and (ordinary or Gaussian) rationals

Some stuff I may have once known but don't quite remember.

I know that the only algebraic integers that are also rational numbers are the ordinary integers. I think the idea is something like, suppose n/m is rational in lowest terms, then if it's a root of a polynomial p(x) with integer coefficients, then p must be divisible by mx−n, and therefore the lead coefficient of p must be divisible by m. Is that right so far? Can you show this just by ordinary long division, or is there some subtlety?

Now, I'm not sure, but I think it's also the case that the only Gaussian rationals that are also algebraic integers are the ordinary integers. Is that true? Can the same argument be made to work? It seems like you would now have a quadratic irreducible polynomial of which your Gaussian rational is a root, and doing long division by that could leave a linear remainder, not just a constant. --Trovatore (talk) 01:23, 15 January 2019 (UTC)

Oh wait. That can't be right, because for example the imaginary unit i is an algebraic integer and a Gaussian rational, but is not an ordinary integer. Is it true that the only Gaussian rationals that are algebraic integers are the Gaussian integers? --Trovatore (talk) 01:25, 15 January 2019 (UTC)

I think you need the leading coefficient of p to be one; otherwise, the argument seems reasonable (if I remember, there is nothing tricky; no abstract algebra is needed). -- Taku (talk) 02:45, 15 January 2019 (UTC)

Also, the answer to the second question should be positive since the ring R of Gaussian integers is integrally closed, meaning a root of a monic polynomial with coefficients in R is in R. -- Taku (talk) 02:50, 15 January 2019 (UTC)

That doesn't sound right. x²+i is a monic polynomial whose coefficients are Gaussian integers, but it has no root that is a Gaussian integer. --Trovatore (talk) 03:00, 15 January 2019 (UTC)

Yes, you're right; I should have said: if x is in Q(i), meaning x is a Gaussian rational (right?), and if x is a root of a monic polynomial with coefficients Gaussian integers, then x is a Gaussian integer (to avoid your counterexample). Showing the ring of Gaussian integers is integrally closed is easy (if you know commutative algebra): it is a PID, thus is a UFD and thus is integrally closed. -- Taku (talk) 04:36, 15 January 2019 (UTC) -- Taku (talk) 04:36, 15 January 2019 (UTC)

I think Gauss's lemma (polynomial) is relevant here. If x is a algebraic integer then the lemma implies the minimum polynomial for x over Z is the same as it's minimum polynomial over Q. The fact that a rational which is also an algebraic integer must be an integer is the special case where the degree of the minimum polynomial is 1. --RDBury (talk) 12:07, 15 January 2019 (UTC)

You can use Gauss’s lemma but it’s even more elementary: suppose $m/n$, m, n relatively prime integers, satisfies the monic polynomial equation $(m/n)^{d}+a_{1}(m/n)^{d-1}+\dots +a_{d}=0$. Now, multiplying the equation by $n^{d}$, we get $-m^{d}=a_{1}m^{d-1}n+\dots +a_{d}n^{d}$. If $n>1$, then this is a contradiction to the assumption $m,n$ are relatively prime. Notice the argument fails if "monic" is dropped and the key is a prime factorization. The same proof also shows a UFD such as the ring of Gaussian integers is integrally closed. —- Taku (talk) 20:14, 15 January 2019 (UTC)

The name rational root theorem is also relevant. --JBL (talk) 23:14, 15 January 2019 (UTC)

Ah, I think I just reproduced the standard proof (but again I just wanted to say it's elementary). I also want to say I don't like the Gauss's lemma (polynomial) article in the current form; I much prefer the French version, which is cleaner and more general. And, the article really should emphasize the application of the lemma to the question such as one here. (Since a UFD is a GCD domain, the idea of proving a version for a GCD itself is not an invalid idea). -- Taku (talk) 23:18, 15 January 2019 (UTC) A new version of Gauss's lemma is now being worked-out at Draft:Gauss's lemma (polynomial). -- Taku (talk) 23:59, 17 January 2019 (UTC)