sage: R.<x> = IntegerModRing(5)[] sage: k = x^3 -3*x + 4 sage: K = k.factor(); K sage: k.is_irreducible()

b. the rationals

sage: R.<x> = CC[] sage: p = x^3 -3*x + 4 sage: P = p.factor(); P sage: p.is_irreducible()

c. the complex numbers

sage: R.<x> = CC[] sage: p = x^3 -3*x + 4 sage: P = p.factor(); P sage: p.is_irreducible()

What are the following commands telling us?

R=Integers()
[R.ideal([a,b]) == R.ideal([gcd(a,b)]) for a in range(1,20) for b in range(1,20)]

Another way to determine if a polynomial is irreducible over a ring, is to check if the zeros of the polynomial are in the ring, ie, does p have a linear factor.

For Z_5, write a "for" loop to check if p evaluated at every element of Z_5 is zero.

Consider the polynomial p=x^3-3*x+4. Use the factor() function to determine if p is irreducible over:

a. finite field Z_5

b. the rationals

c. the complex numbers