Question : Use Eisenstein criterion to show that following polynomials are irreducible in Q[x].
- f(x)=x^5+3x^3-3x+6
- g(x) =x^10+50
Answer :
According to Eisenstein criterion:
If
\[f(x) = a_nx^n + a_{n−1}x^{n−1} + · · · + a_0\]be a polynomial with integer coefficients. Suppose that there exists a prime p which p divides \[a_0 \le i \le n − 1, p \] but p does not divide a_n also p^2 does not divide a_0
Then f(x) is irreducible in Q[x].
- Given polynomial with integer coefficients \[f(x)=x^5+3x^3-3x+6\]
\[=> a_5 =1, a_4 =0, a_3 =3, a_2 =0, a_1 =-3, a_0 =6\]
Since prime number p = 3 divides each of 6,-3,0,3,0 and but 3 does not divide 1 and 9 does not divide 6.
Therefore f(x)=x^5+3x^3-3x+6 is irreducible in Q[x].
2. Given polynomial with integer coefficients \[g(x) =x^10+50\]
\[=> a_10=1, a_9=0, a_8=0,a_7=0,a6=0, a_5 =0, a_4 =0, a_3 =0, a_2 =0, a_1 =0, a_0 =50\]
Choosing a prime number p = 2 Since it divides all \[ a_9=0, a_8=0,a_7=0,a6=0, a_5 =0, a_4 =0, a_3 =0, a_2 =0, a_1 =0, a_0 =50\] but 2 does not divide 1 also 4 does not divide 50 .
Therefore g(x)=x^10+50 is irreducible in Q[x].