Talk:Lagrange's Theorem (Number Theory)

Generalize $\Z_p$ to integral domains?

To cite [1]:

Integral domains have no zero divisors, and consequently allow one to solve equations of

the form $x(y − z) = 0$ in the way one expects: either $x = 0$ or else $y = z$. (This is precisely

the cancellation law for multiplication discussed earlier.

Theorem 1.7. Let $p(X)$ be a polynomial over an integral domain $R$ with degree $n > 0$.

Then the polynomial equation $p(x) = 0$ has at most $n = \deg(p(X))$ roots in $R$.

The proof is the same as Lagrange's Theorem (Number Theory) for $\Z_p$. --Hbghlyj (talk) 23:37, 17 March 2024 (UTC)

Feel free to write another page for this. Please do not amend the statement of the theorem of this page, of course, for obvious reasons. --prime mover (talk) 20:25, 18 March 2024 (UTC)

Though the proof is almost the same, it will need Polynomial Factor Theorem for integral domains. But the current Polynomial Factor Theorem is only for fields. So we need to write another page of Polynomial Factor Theorem for integral domains first. --Hbghlyj (talk) 21:54, 18 March 2024 (UTC)

That is how this site works. One finds the dissertation of what one wants to write about in the literature. One then uses that source work by building up the material from first principles for that work.

Then, when one has worked through the book to the level of the result in it, that piece of work is ready to be presented.

It has been shown to be a much more difficult endeavour to start at the top and work down. It is vital for the philosophical integrity of $\mathsf{Pr} \infty \mathsf{fWiki}$ that the system of interlinking that we are so rigorously implementing is as correct as possible. With a large workforce of contributors, ensuring consistency and quality is mainly upon the contributors themselves, but when two or more contributors (some of whom are no longer around) have been working in the same field, insuring consistency of notation, terminology and direction can indeed be challenging.

Hence the stress placed upon accuracy of definitions and validity of links.

Sorry about my intolerance of StackExchange. It's just that questions and answers there are not guaranteed to be eternal. There is a small handful of online resources which are trusted, the main one being MathWorld, which is excellent, but doesn't give proofs and is surprisingly deficient in some areas.

With that in mind, go to with a will, but please be prepared to have your work challenged and amended if this is necessary for the philosophy of $\mathsf{Pr} \infty \mathsf{fWiki}$.