Name:
Problem Set 9
Math 415 Honors, Fall 2014
Due: Tuesday, November 25.
Review Sections 20, 21, 22 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Tuesday, November 25. Remember to fully justify all your answers,
and provide complete details. Neatness is greatly appreciated.
1. Find all solutions x ∈ Z of the congruence equation 39x ≡ 52 mod 130.
2. Compute the remainder of 3749 upon division by 7. Compute the remainder of 71000 upon
division by 24.
3. Let p, q be distinct prime numbers. Compute ϕ(p2 ) and ϕ(pq), where ϕ is Euler’s function.
4. Let p be a prime number. Show that, in Z p , the equation x2 = 1 has exactly two solutions
x = 1, x = p − 1.
5. Use the previous exercise to show that if p is prime, (p − 1)! ≡ −1 mod p.
6. Describe (as a subfield of the complex numbers
C) the field of fractions of the integral domain
√
D = {n + mi | n, m ∈ Z}. Here, i = −1. You do not need to check that D is an integral
domain.
7. List all polynomials of degree 2 in Z5 [x]. How many are there?
8. Consider the evaluation homomorphism φ3 : Z5 [x] → Z5 given by φ3 ( f (x)) = f (3). Compute
φ3 (x231 + 3x117 − 2x53 + 1).
9. Find all zeros of 2x219 + 3x74 + 2x57 + 3x44 in Z5 .
10. Let D be an integral domain. Show that D[x] is also an integral domain.
11. Let F be a field of characteristic zero. We define formal differentiation on F[x] via
(an xn + an−1 xn−1 + · · · + a1 x + a0 )0 = nan xn−1 + (n − 1)an−1 xn−2 + · · · + a1
Show that 0 : F[x] → F[x] is a group homomorphism of the additive group (F[x], +). Show
that this is not a ring homomorphism. Find the kernel and the image of the formal differentiation map.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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