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$\mathrm{ord}_{p}(a+b)\ge\mathrm{min}(\mathrm{ord}_{p}a,\mathrm{ord}_{p}b)$ with equality holding if $\mathrm{ord}_{p}a\ne \mathrm{ord}_{p}b$. is a the statement that prompted this question.

It was found in Ireland & Rosen's Elements of Number Theory (precurser to their book A Classical Introduction to Modern Number Theory) is the book that I am working through and it asks for its proof.

After some research I'm learned that this function is completely additive($\mathrm{ord}_{p}(ab)=\mathrm{ord}_{p}(a)+\mathrm{ord}_{p}(b)$) among other things and managed to get the following equation out that if I didn't make a mistake, solves the first part of the question:

$\mathrm{ord}_{p}(a+b)=\mathrm{ord}_{p}(dm+dn)=\mathrm{ord}_{p}(d)+\mathrm{ord}_{p}(m+n)\ge \mathrm{ord}_{p}(d)=\mathrm{ord}_{p}(a,b)=\mathrm{min}(\mathrm{ord}_{p}a,\mathrm{ord}_{p}b)$

Where $(a,b)$=$\mathrm{gcd}(a,b)$ is the ideal/greatest common factor and is equal to $d$, and $a=dm$, $b=dn$ where $m$,$n$ are relatively prime.


So my question is where can I learn more about the properties and identities of the functions $\mathrm{ord}_{p}(n)$ and $\mathrm{v}(n)$ (the latter seems to be called valuation or related to such), especially a resource that would include relations similar to those above. If the resource included identities for the lcm, gcd, and min/max functions in relation to the ord function and by themeslves also, that would be wonderful.

Also what are some identities that could solve the "with equality holding if $\mathrm{ord}_{p}a\ne \mathrm{ord}_{p}b$." part and how are they derived? Edit(This part has been answered, looking for some good resources that expound on ord,v, and related functions.)

Thank you for your assistance.

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3 Answers 3

up vote 1 down vote accepted

Gerry's answer is great, though I can't help but add that this is a deep and beautiful idea that generalises and is extremely useful in geometric situations as well (algebraic geometry and commutative algebra). See here, for example.

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I'm finding this "Valuation (Algebra)" topic very interesting and the article has led to some great additional resources, just wanted to thank you personally. –  gmlime Jul 31 '12 at 10:29

I think what you need to know is that given any non-zero integer $a$ and any prime $p$ you have $$a=p^rm$$ for some non-negative integer $r$ and some integer $m$ with $\gcd(m,p)=1$, and that $r$ and $m$ are unique. This falls out of the Unique Factorization Theorem. Then $$ord_p(a)=r$$ You should be able to use this to prove the results you want on $ord_p(a+b)$.

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You can push your method of proof further to get the equality result.

Suppose that $ord_p(a) \neq ord_p(b)$. Then one of $m$ or $n$ is divisible by $p$, but the other isn't. (Check using the definitions!) Thus $m +n$ is not divisible by $p$, so $ord_p(m+n) = 0$. This gives what you want.

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That answers the question that originally got me thinking about this. I wish I could accept more than one response. Thanks. –  gmlime Jul 31 '12 at 7:03

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