For how many positive integers 1≤k≤1000 is the polynomial fk(x)=x^3+x+k irreducible?
To determine the number of positive integers 1≤k≤1000 for which the polynomial fk(x)=x^3+x+k is irreducible, we need to consider the Eisenstein's criterion for irreducibility.
Eisenstein's criterion states that if a polynomial has integer coefficients and satisfies the following conditions:
1. The leading coefficient is not divisible by any prime p.
2. All other coefficients except the constant term are divisible by p.
3. The constant term is not divisible by p^2.
Then, the polynomial is irreducible.
In our case, the polynomial fk(x)=x^3+x+k has the leading coefficient 1, so it satisfies the first condition.
Now, let's consider the second condition. We need to choose a prime p such that all coefficients except the constant term (k) are divisible by p. Notice that the coefficients of the polynomial are either 1 or 0. So, if we choose any prime p, it will always divide the coefficients except the constant term.
Finally, we consider the third condition. The constant term (k) should not be divisible by p^2 for the polynomial to be irreducible.
Since we want positive integers k, we need to find the prime numbers p for which the constant term (k) is not divisible by p^2.
There are 168 prime numbers between 1 and 1000. We can check each of these primes, p, and count how many corresponding values of k satisfy the condition that k is not divisible by p^2.
Therefore, the number of positive integers 1≤k≤1000 for which the polynomial fk(x)=x^3+x+k is irreducible can be determined by checking each prime number p and counting the corresponding values of k that satisfy the condition.