Question #146088

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.


1
Expert's answer
2020-11-24T17:33:48-0500

Let us show that the polynomial 3x5+15x4+20x3+10x+203x^5 + 15x^4+20x^3 + 10x + 20 is irreducible over Q\mathbb Q.


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

p(x)=anxn+an1xn1++a1x+a0{\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 ai for 0 ≤ i < n,
  • p does not divide an, and
  • p2 does not divide a0,

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


In our case, let p=5p=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, 3x5+15x4+20x3+10x+203x^5 + 15x^4+20x^3 + 10x + 20 is irreducible over Q\mathbb Q.



Is the polynomial 3x2+x+43x^2 + x + 4 irreducible over Z7\mathbb Z_7 ?


Since 3x2+x+4=(x+2)(3x+2)3x^2 + x + 4=(x+2)(3x+2), we conclude that this polynomial is reducible over Z7\mathbb Z_7 (note that 8 = 1 in the field Z7\mathbb Z_7).



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