Factoring Polynomials over Various Rings

​
a
4
1
a
3
0
a
2
0
a
1
0
a
0
1
number system

p

[i]
p
2
In [i]
1+
4
x
(
2
x
-)(
2
x
+)
This Demonstration looks at the factorization of polynomials over the integers, the Gaussian integers, and the finite fields

p
, where
p
is prime.
Irreducible polynomials are used to generate new fields using the ring of cosets formed by the ideal generated by the irreducible polynomial. Polynomials of degree two or three are irreducible if and only if they have no linear factors and hence no roots. Polynomials of degree four are thus the first interesting case. If there is a prime
p
such that the polynomial
q(x)
is irreducible in the finite field

p
, then
q(x)
is irreducible over the integers.
However, there are irreducible polynomials over the integers that factor over all primes. The Gaussian integers
[]
are an alternative number system used to check for factorization.

Details

Snapshot 1:
4
x
+1
does factor over the Gaussian integers using
i=
-1
Snapshot 2:
4
x
+1
factors into four linear factors in

17
[x]
Snapshot 3:
4
x
+1
factors into two quadratic factors in

3
[x]
J. A. Gallian, Contemporary Abstract Algebra, 7th ed., Belmont, CA: Brooks/Cole, 2010.

External Links

Finite Field (Wolfram MathWorld)
Integer Polynomial (Wolfram MathWorld)

Permanent Citation

Theodore S. Erickson, Eric W. Weisstein
​
​"Factoring Polynomials over Various Rings"​
​http://demonstrations.wolfram.com/FactoringPolynomialsOverVariousRings/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011