It is proved that the polynomial f (x) = x ^ 3 + 3x + 1 is irreducible in the field of rational numbers College Advanced Algebra for help!

It is proved that the polynomial f (x) = x ^ 3 + 3x + 1 is irreducible in the field of rational numbers College Advanced Algebra for help!

If a polynomial of degree 3 is reducible in the field of rational numbers, it must contain rational factors of degree 1
In other words, there must be a rationale
Let f (x) have rational roots P / Q, where P and Q are coprime integers
As a polynomial with integral coefficients, f (x) can prove that P divides the constant term and Q divides the first term
For f (x) = x ^ 3 + 3x + 1, only P / Q = 1 or - 1
But it is easy to verify that both 1 and - 1 are not roots of F (x), so f (x) has no rational root, so it is irreducible in rational number field
Note that for polynomials with rational coefficients of degree 4 or more,
The absence of rational roots is only a necessary and not sufficient condition for irreducibility over rational fields