# inverse of polynomial in division ring

#### skyer

Hello,
I'm supposed to find inverse to the polynomial:
$x^2+1+(f)$ in $Q[x]/((f),+,*)$
and $f=x^3+5$.
I know the solution somehow involves extended Euclidean algorithm, but generally don't know what to do. I tried to find the coefficients for Bezout identity which worked, but don't know how to continue.

Can somebody help me with this?

#### Idea

yes, use the Euclidean algorithm

$$\displaystyle x^3+5=\left(x^2+1\right)x +(5-x)$$

$$\displaystyle x^2+1=(5-x)(-x-5)+26$$

Therefore

$$\displaystyle \left(x^2+1\right)\left(\frac{1}{26}-\frac{5}{26}x-\frac{1}{26}x^2\right)+$$

$$\displaystyle \left(x^3+5\right)\left(\frac{x}{26}+\frac{5}{26}) =1$$

so the inverse is

$$\displaystyle \frac{1}{26}-\frac{5}{26}x-\frac{1}{26}x^2$$

Last edited:

#### skyer

The textbook states that this is the way we verify it - $(x^2+1)*inverse=1 \mod f$ - and it works... but why?

I don' really understand the notation - We want an inverse to $x^2+1+(f)$ but we verify correctness with formula using multiplication. Would you point me in the right direction? (wiki link is more than sufficient. thanks.)

#### Idea

The textbook states that this is the way we verify it - $(x^2+1)*inverse=1 \mod f$ - and it works... but why?

I don' really understand the notation - We want an inverse to $x^2+1+(f)$ but we verify correctness with formula using multiplication. Would you point me in the right direction? (wiki link is more than sufficient. thanks.)
Q[x] / (f) is a field, (f) is the ideal generated by f

a typical element is written in the form h(x) + (f) where h(x) is in Q[x]

the 0 element is written as 0 +(f) = (f)

the multiplicative identity 1 + (f)

so the inverse of h(x) would be g(x) such that ( h(x) + (f) ) * ( g(x) + (f) ) = 1 + (f)

which is the same as h(x)*g(x) + (f) = 1 + (f)

This is why we use multiplication to verify the answer.

Is this what you are asking?

#### skyer

Yes, thank you.

This would work with other fields too, right? eg. if we used Z_prime instead of Q?

#### Idea

sure.

you need an irreducible polynomial f in $$\displaystyle \mathbb{Z}_p[x]$$

to make sure that the quotient is a field so every nonzero element has an inverse

#### skyer

Great,
thanks a lot for help. I have another question from algebra that I'll post soon, I'd be really grateful if you helped me with that one too.