Question

Q: Would like to clearly understand and solve this example from Modern Algebra II with the steps of the solution to better understand, thanks.

**Please explain and give the step by step with details to completely see how the solution came about, plenty of thanks.

1) Determine which of the polynomials below is or are irreducible over Q.

a) x^4 + 3x^2 + 3.

b) x^5 + 5x^2 + 1.

Answer 1

Eisenstein Criterion :

Suppose we have the following polynomial with integer coefficients.

If there exists a prime number p such that the following three conditions all apply:

• p divides each a_i for i ≠ n,
• p does not divide a_n, and
• p² does not divide a₀

Then Q(x) is irreducible over the rational number .

(a) . x⁴ + 3x² + 3

consider p=3 then ,

p does not divides 1 , p divides 3 and p² does not divides 3 . Hence by Eisenstein croterion x⁴ + 3x² + 3 is irreducible over Q .

(b) . x⁵ + 5x² +1

Since both the constant term and leadind coefficient is 1 so the polynomial have rational root 1 or -1 but 1, -1 are not root of the polynomial . So if it is reducible it can be written as two product of factors one of them have degree 2 and other one have 3 .

uppose Cat Equat nboth Side both Side a ConAadie4ion 3 T2 o be written as gvで ら วั t 5a t l is irreducible. nce

.

If you have any doubt or need more clarification at any step please comment.