# irreducibility using factor function()

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

# c. the complex numbers

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This looks like homework. Please always mention the own effort to make sage compute something on the way. Here, a possibility to proceed - for each field - would be to construct the corresponding field, the ring of polynomials over it, then define $p$ over this field, finally ask for the factorization. For instance:

sage: F = GF(5)
sage: F.<x> = PolynomialRing( F )    # or F[] for short, but criptic.
sage: factor( x^3 - 3*x + 4 )
x^3 + 2*x + 4


Which is the code for the other two fields? (Of course, we can already decide...)

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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()

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