# Talk:Formal derivative

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Field:  Algebra

The slick alternative definition of the formal derivative, using [ f(Y)-f(X) ] / [Y - X], is due to Bill Dubuque of MIT, if I'm not mistaken. LDH (talk) 04:51, 26 November 2008 (UTC)

## Repeated factors or roots

Might it be simpler to say that formal derivatives can be used to detect repeated factors rather then repeated roots? It would save introducing the notions of extensions and closures, and the example would become ${\displaystyle x^{6}+1=(x^{2}+1)^{3}{\pmod {3}}}$. Also, if the example had factors with different multiplicities, then it could be mentioned that these could be separated out by repeatedly taking the GCD. --catslash (talk) 11:27, 20 July 2009 (UTC)

I agree with Catslash.Hippo.69 (talk) 09:35, 31 December 2012 (UTC)

## Noncomutative rings?

How does the polynomials look like for noncomutative rings ... would not ${\displaystyle \sum _{i=0}^{n}a_{i}x^{i}b_{i}}$ be the standard form? Hippo.69 (talk) 09:35, 31 December 2012 (UTC)

It seems to me, polynomials over noncomutative rings could not be described by easy standard form ... either it is not closed on addition or it is difficult to test that two polynomials are identical. ... Possibility closed on addition would be to maintain list of pairs (a,b) for each power. One could shorten the lists when two pairs with the same a or with the same b are found ... but that could lead to different (list of) lists for the same polynomial.Hippo.69 (talk) 11:25, 2 January 2013 (UTC)

## Better definition for noncomutative rings

I have used for Czech version a'=0 for scalars and x'=1 for atomic terms and for composed terms (a+b)'=a'+b' and (ab)'=a'b+ab' as the definition. One just should prove the derivative gives the same result for terms which are equal according to equations from ring axioms (commutativity for addition, associativity for both addition and multiplication and distributivities). The derivative for polynomial is an easy consequence. Hippo.69 (talk) 22:51, 1 January 2013 (UTC)

I have finally decided to write this to the main article Hippo.69 (talk) 19:39, 1 April 2013 (UTC)