Is there a faster way to do this?

[e12514]

Z/p [x] consists of polynomials in x with coefficients in a field Z/p.
I is an ideal of Z/p [x] generated by some irreducible polynomial i(x) in Z/p [x]
(let's say I is generated by an irreducible polynomial i(x) of degree 3 in Z/p [x])

The factor (or quotient)
(Z/p [x])/I = (Z/p [x])/<i(x)> is a field.
To verify that it is,
* (Z/p [x])/<i(x)> is commutative because Z/p [x] is. (ok)
* i(x) is irreducible so (Z/p [x])/<i(x)> has no divisors of zero. (ok)
* (Z/p [x])/<i(x)> needs to be a division ring, that is, every nonzero element needs to be invertible. (needs to be verified)

My (tedious) method is,
WLOG let f(x) = x^2 + ax + b be a polynomial in Z/p [x]
need to find some iverse of f(x) (call it g(x)) in Z/p [x] such that
(f(x).g(x)) + <i(x)> = 1 + <i(x)>

which can be done by multiplying out f(x).g(x) and then dividing by i(x) to get the remainder (very long) which would be of degree 2 (one less than deg(i(x)) )
and then solving the coefficient of x^2 and x to be congruent to 0 mod p
and the constant term to be congruent to 1 mod p
(three equations and three unkowns)

and not only does that take very long, if p is not small then in the end I often end up making a mistake somewhere so my calculated "inverse" of f(x) is incorrect
(end up having f(x).g(x) + <i(x)> different to 1 + <i(x)> )

Is there a better/cleaner/easier/more efficient way to do these sorts of things rather than having to go through all that mess?
(Can take p to be a specific prime, say p=5, or 7, or whatever.)

 

[morphism]

Z/p is a field, so (Z/p)[x] is a principal ideal domain. If i(x) is irreducible over Z/p, then <i(x)> is a prime ideal and consequently (because we're in a PID) a maximal ideal. And what happens when you mod out by a maximal ideal?

 

[e12514]

Then the factor is a field! Yay, fantastic! Thanks!
This shows that I need to do a lot of extra reading on stuff that is not taught in order to fully understand the material.. I've never heard of a pid before, unless it's lying on some fresh page that I've never looked at in our notes, or it's not covered at my level - first course in abstract algebra... (I had to look up maximal and prime ideals too...)