Answer to Question #146088 in Abstract Algebra for Sourav Mondal

Question

Show that the polynomial

3x^(5) + 15x⁴+20x³ + 10x + 20 is irreducible over Q. Is the polynomial 3x² + x + 4 irreducible over Z7 ? Give

reasons.

Answer 1

Let us show that the polynomial $3x^5 + 15x^4+20x^3 + 10x + 20$ is irreducible over $\mathbb Q$.

We use Eisenstein's criterion. Suppose we have the following polynomial with integer coefficient:

${\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}$

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

• p divides each a_i for 0 ≤ i < n,
• p does not divide a_n, and
• p² does not divide a₀,

then $p(x)$ is irreducible over the rational numbers.

In our case, let $p=5$.

Then 5 | 15, 5 | 20, 5 | 10, 5 | 20.

Also 5 does not divide 3, and

25 does not divide 20.

Therefore, according to Eisenstein's criterion, $3x^5 + 15x^4+20x^3 + 10x + 20$ is irreducible over $\mathbb Q$.

Is the polynomial $3x^2 + x + 4$ irreducible over $\mathbb Z_7$ ?

Since $3x^2 + x + 4=(x+2)(3x+2)$, we conclude that this polynomial is reducible over $\mathbb Z_7$ (note that 8 = 1 in the field $\mathbb Z_7$).