Non commutative ring and polynomial faggotry

Howdy, chaps ?

Say, has any one really been far even as to do look more like ?

Consider a polynom with coefficients over a non-commutative ring. How do you define roots of such a polynom ? Basicly, how do you define the induced factorisation of such a polynom ? What can you tell about the aforementionned root (numbers, relations, sauce) ?

And the least important part to us mathematicians : is this useful for anything ?

>>8

At first I did'nt think of "left" or "right" thingies. I wondered how you could obtain a root factorisation when using non-commutative coefficients since the order matters. It's clear now, thanks to you. The last thing I need is if there is a property on the numbers or the multiplicity of roots.

The whole problem appeared when I was studying some algebra book. The author starts by showing that the multiplicative group of a finite field is always cyclic, but he's using a commutative property to do so. Atm he has not demonstrate the Wedderburn theorem so we must assume the fiel is not commutative, hence faggotized his demonstration. The main point was : X^d-1 has d or less roots in a field. So in non-commutative business this is wrong and I was fucked up.