Question

(a) For an odd prime $p$, prove that the congruence $2x^2 + 1 = 0 \in \mathbb{Z}_p$ has a solution if and only if $p = 1, 3 \in \mathbb{Z}_8$. (b) Solve the congruence $2x^2 + 1 = 0 \in \mathbb{Z}_{121}$.

          (a) For an odd prime $p$, prove that the congruence $2x^2 + 1 = 0 \in \mathbb{Z}_p$ has a solution if and only if $p = 1, 3 \in \mathbb{Z}_8$.
(b) Solve the congruence $2x^2 + 1 = 0 \in \mathbb{Z}_{121}$.

(a) For an odd prime p, prove that the congruence 2x^2 + 1 = 0 ∈ℤp has a solution if and only if p = 1, 3 ∈ℤ8.
(b) Solve the congruence 2x^2 + 1 = 0 ∈ℤ121.

Added by Anne E.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert James Chok

11/25/2019

Step 1

We want to show that this congruence has a solution if and only if p = 1 or p = 3 (mod 8). Show more…

Show all steps

Thanks for your feedback!

(a) For an odd prime p, prove that the congruence 2x^(2)+1=0inZ_(p) has a solution if and only if p=1,3inZ_(8). (b) Solve the congruence 2x^(2)+1=0inZ_(121). (a) For an odd prime p, prove that the congruence 2x2 + 1 = 0 e Zp has a solution if and only if p = 1,3 e Zs (b) Solve the congruence 2x2 + 1 = 0 e Z121.