Question

Let f ∈ Z[x] be a nonconstant polynomial. Prove that the set S = {p prime:...

Let f ∈ Z[x] be a nonconstant polynomial. Prove that the set S = {p prime: there exist infinitely many positive integers n such that p | f(n)} is infinite.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let f ∈ Z[x] be a nonconstant polynomial with the property that all the roots (in...
Let f ∈ Z[x] be a nonconstant polynomial with the property that all the roots (in comlex plane) for the equation f(x) = 0 are distinct. Prove that there exist infinitely many positive integers n such that f(n) is not a perfect square.
Let f ∈ Z[x] be a nonconstant polynomial with the property that all the roots (in...
Let f ∈ Z[x] be a nonconstant polynomial with the property that all the roots (in comlex plane) for the equation f(x) = 0 are distinct. Prove that there exist infinitely many positive integers n such that f(n) is not a perfect square. Could you explain it in number theory instead of some deep math like sigel theorem
Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree...
Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D_2p of a regular p-gon. Prove that f (x) has either all real roots or precisely one real root.
Let p be a prime and m an integer. Suppose that the polynomial f(x) = x^4+mx+p...
Let p be a prime and m an integer. Suppose that the polynomial f(x) = x^4+mx+p is reducible over Q. Show that if f(x) has no zeros in Q, then p = 3.
Let f: Z→Z be the functon defined by f(x)=x+1. Prove that f is a permutation of...
Let f: Z→Z be the functon defined by f(x)=x+1. Prove that f is a permutation of the set of integers. Let g be the permutation (1 2 4 8 16 32). Compute fgf−1.
For each polynomial f(x) ∈ Z[x], let f ' (x) denote its derivative, which is also...
For each polynomial f(x) ∈ Z[x], let f ' (x) denote its derivative, which is also a polynomial in Z[x]. Let R be the following subset of Z[x]: R = {f(x) ∈ Z[x] | f ' (0) = 0}. (a) Prove that R is a subring of Z[x]. (b) Prove that R is not an ideal of Z[x].
Let a1, a2, ..., an be distinct n (≥ 2) integers. Consider the polynomial f(x) =...
Let a1, a2, ..., an be distinct n (≥ 2) integers. Consider the polynomial f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x] (1) Prove that if then f(x) = g(x)h(x) for some g(x), h(x) ∈ Z[x], g(ai) + h(ai) = 0 for all i = 1, 2, ..., n (2) Prove that f(x) is irreducible over Q
Let p be a prime. (a) Prove that Z/pZ ⊕ Z/pZ has exactly p + 1...
Let p be a prime. (a) Prove that Z/pZ ⊕ Z/pZ has exactly p + 1 subgroups of order p. (b) How many subgroups of order p does Z/pZ ⊕ Z/pZ ⊕ Z/pZ have? Can you generalize further? Explain.
1. Let Z[i] denote the set of all ‘complex numbers with integer coefficients’:the set of all...
1. Let Z[i] denote the set of all ‘complex numbers with integer coefficients’:the set of all a + bi such that a and b are integers. We say that z is composite if there exist two complex integers v and w such that z=vw and |v|>1 and |w|>1. Then z is prime if it is not composite A) Prove that every complex integer z, |z| > 1, can be expressed as a product of prime complex integers.
Let F be a field and let a(x), b(x) be polynomials in F[x]. Let S be...
Let F be a field and let a(x), b(x) be polynomials in F[x]. Let S be the set of all linear combinations of a(x) and b(x). Let d(x) be the monic polynomial of smallest degree in S. Prove that d(x) divides a(x).
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT