Question

4. (15 points) For this problem, you may take for granted that 47 and 53 are prime, and that 2491 = 47 \cdot 53. Consider the polynomial $f = X^2 + 3$. Justify your answer for each part. (a) Determine whether or not the polynomial $X^2 + 3$ has a root in $F_{47}$. (b) Determine whether or not the polynomial $X^2 + 3$ has a root in $F_{53}$. (c) Determine whether or not the polynomial $X^2 + 3$ has a root in $\mathbb{Z}/2491\mathbb{Z}$.

          4. (15 points) For this problem, you may take for granted that 47 and 53 are prime, and that 2491 = 47 \cdot 53. Consider the polynomial $f = X^2 + 3$. Justify your answer for each part.
(a) Determine whether or not the polynomial $X^2 + 3$ has a root in $F_{47}$.
(b) Determine whether or not the polynomial $X^2 + 3$ has a root in $F_{53}$.
(c) Determine whether or not the polynomial $X^2 + 3$ has a root in $\mathbb{Z}/2491\mathbb{Z}$.

4. (15 points) For this problem, you may take for granted that 47 and 53 are prime, and that 2491 = 47 ·53. Consider the polynomial f = X^2 + 3. Justify your answer for each part.
(a) Determine whether or not the polynomial X^2 + 3 has a root in F47.
(b) Determine whether or not the polynomial X^2 + 3 has a root in F53.
(c) Determine whether or not the polynomial X^2 + 3 has a root in ℤ/2491ℤ.

Added by Stephanie G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Lucas Gagne

10/06/2020

Step 1

Step 1: To determine whether or not the polynomial X^2 + 3 has a root in F47, we need to check if there exists an element a in F47 such that a^2 + 3 = 0. Show more…

Show all steps

Thanks for your feedback!

4. (15 points) For this problem, you may take for granted that 47 and 53 are prime, and that 2491 = 47. 53. Consider the polynomial f = X2 + 3. Justify your answer for each part. (a) Determine whether or not the polynomial X2 + 3 has a root in F47. (b) Determine whether or not the polynomial X2 + 3 has a root in F53. (c) Determine whether or not the polynomial X2 + 3 has a root in Z/2491Z