Question

Theorem 16.15. Suppose p is a prime number satisfying p ? 1 (mod 4). 1. The equation x² + y² = p has integer solutions, p factors nontrivially in ?[i], the polynomial x² + 1 factors nontrivially in ??[x], and ?1 is a square in ??. 2. There are eight solutions to the equation x² + y² = p. Each solution (a, b) corresponds to a pair of irreducible Gaussian integers a + bi and a ? bi such that p = (a + bi)(a ? bi). Exercise 16.12. Prove Theorem 16.15.

          Theorem 16.15. Suppose p is a prime number satisfying p ? 1 (mod 4).

1. The equation x² + y² = p has integer solutions, p factors nontrivially in ?[i], the polynomial x² + 1 factors nontrivially in ??[x], and ?1 is a square in ??.

2. There are eight solutions to the equation x² + y² = p. Each solution (a, b) corresponds to a pair of irreducible Gaussian integers a + bi and a ? bi such that p = (a + bi)(a ? bi).

Exercise 16.12. Prove Theorem 16.15.

Theorem 16.15. Suppose p is a prime number satisfying p ? 1 (mod 4).

1. The equation x² + y² = p has integer solutions, p factors nontrivially in ?[i], the polynomial x² + 1 factors nontrivially in ??[x], and ?1 is a square in ??.

2. There are eight solutions to the equation x² + y² = p. Each solution (a, b) corresponds to a pair of irreducible Gaussian integers a + bi and a ? bi such that p = (a + bi)(a ? bi).

Exercise 16.12. Prove Theorem 16.15.

Added by William N.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Sri K

03/23/2023

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Theorem 16.15. Suppose p is a prime number satisfying p ≡ 1 (mod 4). 1. The equation x² + y² = p has integer solutions, p factors nontrivially in ℤ[i], the polynomial x² + 1 factors nontrivially in ဴp[x], and -1 is a square in ဴp. 2. There are eight solutions to the equation x² + y² = p. Each solution (a, b) corresponds to a pair of irreducible Gaussian integers a + bi and a - bi such that p = (a + bi)(a - bi). Exercise 16.12. Prove Theorem 16.15.