Question

5. (a) Suppose p is prime, p ? a and k = gcd(n, p - 1). If a^{(p-1)/k} ? 1 (mod p), prove that the congruence x^n ? a (mod p) has a solution. (b) For a prime p and p ? a, a is called a cubic residue modulo p if the congruence x^3 ? a (mod p) has a solution. Prove that if p = 3k + 2 then all the integers in a reduced residue system are cubic residues modulo p.

          5. (a) Suppose p is prime, p ? a and k = gcd(n, p - 1). If
a^{(p-1)/k} ? 1 (mod p),
prove that the congruence x^n ? a (mod p) has a solution.
(b) For a prime p and p ? a, a is called a cubic residue modulo p if the congruence
x^3 ? a (mod p)
has a solution. Prove that if p = 3k + 2 then all the integers in a reduced residue system are cubic residues modulo p.

5. (a) Suppose p is prime, p ? a and k = gcd(n, p - 1). If
a^(p-1)/k ? 1 (mod p),
prove that the congruence x^n ? a (mod p) has a solution.
(b) For a prime p and p ? a, a is called a cubic residue modulo p if the congruence
x^3 ? a (mod p)
has a solution. Prove that if p = 3k + 2 then all the integers in a reduced residue system are cubic residues modulo p.

Added by Mark J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Maitreya E

Step 1

We are also given that 4 ≡ 1 (mod p). We want to prove that the congruence z^n ≡ a (mod p) has a solution. Since p is prime and p does not divide a, we can use Fermat's Little Theorem, which states that a^(p-1) ≡ 1 (mod p). Now, since k = gcd(n, p-1), we can Show more…

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5. (a) Suppose p is prime, p ∤ a and k = gcd(n, p - 1). If a^{(p-1)/k} ≡ 1 (mod p), prove that the congruence x^n ≡ a (mod p) has a solution. (b) For a prime p and p ∤ a, a is called a cubic residue modulo p if the congruence x^3 ≡ a (mod p) has a solution. Prove that if p = 3k + 2 then all the integers in a reduced residue system are cubic residues modulo p.