Let $f(x)=x^7-105x+12.$ Show that $f(m)$ is not prime for any integer $m$.
Asked
Active
Viewed 42 times
0
-
Can you factor the polynomial? – user247327 Jun 15 '19 at 12:31
-
1Member for 2 years and 11 months and You don't write about Your own attempts? Did You read this? – Peter Melech Jun 15 '19 at 12:31
-
@user247327 The polynomial $f$ is irreducible over $\Bbb Q$, so there is no non-trivial factorisation. – Dietrich Burde Jun 15 '19 at 12:32
-
@user 247327 which test you're using to conclude that f(x) is irreducible over $Q$ – Ravindra Jun 15 '19 at 13:29
-
@Ravi Eisenstein with $p=3$. – Dietrich Burde Jun 15 '19 at 15:01
-
@Dietrich Burde $f$ is irreducible over $Q$. From this how can we conclude that $f(m)$ is not prime for any integer $m$. – Ravindra Jun 15 '19 at 15:44
-
@Ravi We cannot conclude this. Take $f=x^2+x+1$. It is irreducible, but $f(1)$ is prime. – Dietrich Burde Jun 15 '19 at 18:07
-
@Dietrich Burde I had the same doubt. – Ravindra Jun 16 '19 at 12:33
1 Answers
4
We have $2\mid f(x)$ for all integers $x$. On the other hand, $f(x)=2$ is impossible over the integers, since $x^7-105x+10$ has no rational root by the rational root theorem.
Dietrich Burde
- 130,978