Math 3527 ~ Number Theory 1, Spring 2024 ~ Homework 8, due Tue Mar 26th.
Part I: No justi cations are required for these problems. Answers will be graded on correctness.
1. Let R = F3 [x] and p = x2 + x.
(a) List the 9 residue classes in R/pR. (You may omit the bars in the residue class notation.)
(b) Construct the addition and multiplication tables for R/pR.
(c) Identify all of the units and zero divisors in R/pR.
(d) Find the order of each unit in R/pR. Are there any primitive roots?
(e) Verify Euler's theorem for each unit in R/pR.
2. Let R = F2 [x] and p = x3 + x + 1.
(a) List the 8 residue classes in R/pR. (You may omit the bars in the residue class notation.)
(b) Construct the addition and multiplication tables for R/pR.
(c) Show that R/pR is a eld by explicitly identifying the inverse of every nonzero element. [Hint: Use the multiplication table from (b).]
(d) Find the order of each unit in R/pR. Are there any primitive roots?
(e) Verify Fermat's little theorem for the elements x and x + 1 in R/pR.
3. Let R = Z[i] and p = 2 + 2i. You are given that there are 8 residue classes modulo p, represented by 0, 1, 2, -1, 1 - i, i, 1 + i, and -i.
(a) Construct the addition and multiplication tables for R/pR. (Please leave the elements in the order given above: when you work out the tables you will see they are given in that order for a reason!)
(b) Identify all of the units and zero divisors in R/pR.
(c) Find the order of each unit in R/pR. Are there any primitive roots?
4. Find the following multiplicative inverses:
(a) The multiplicative inverse of x + 3 inside Q[x] modulo x2 + 1.
(b) The multiplicative inverse of 1 - 2i inside Z[i] modulo 8 + 7i.
(c) The multiplicative inverse of x2 + 1 inside F3 [x] modulo x4 + 2x + 1.
(d) The multiplicative inverse of 4 + 8i inside Z[i] modulo 11 - 14i.
5. (a) Solve the simultaneous congruences p 三 1 (mod x + 2) and p 三 7 (mod x - 1) in Q[x].
(b) Solve the simultaneous congruences z 三 1 (mod 2 + 2i) and z 三 -i (mod 4 + 5i) in Z[i].
Part II: Solve the following problems. Justify all answers with rigorous, clear explanations.
6. Show the following things:
(a) Show that the element 4 + 5i is irreducible and prime in Z[i].
(b) Show that the element x2 + 4x + 5 is irreducible and prime in R[x].
(c) Show that the element x2 + 4x + 5 is neither irreducible nor prime in C[x] by nding a factorization.
(d) Show that the element 3 + 5i is neither irreducible nor prime in Z[i] by nding a factorization.
(e) Show that the element 2 + √-10 is irreducible but not prime in Z[√-10]. [Hint: Show it divides 14 and that there are no elements of norm 2 or 7.]
7. We can use successive squaring and the same order-calculation procedure we used in Z/mZ to establish the order of an arbitrary unit residue class s in R/rR: explicitly, s has order n if and only if sn = 1 and sn/p ≠ 1 for any integer prime p dividing n.
(a) Show that the element 2 + i has order 8 in Z[i] modulo r = 3 + 5i.
(b) Show that the element x has order 6 in F7 [x] modulo r = x2 + x + 5.
(c) Show that R = F5 [x] modulo r = x2 + 2 is a eld with 25 elements, and deduce that the order of any nonzero residue class in R/rR divides 24.
(d) Find the orders of 2, x, and x + 1 in F5 [x] modulo x2 + 2. Are any of them primitive roots? [Hint: By
(c), the order of each element divides 24, so search among divisors of 24.]
(e) Show that R = F5 [x] modulo p = x2 is not a eld, and in fact that there are 20 units in R/rR.
(f) Find the orders of 2, x + 1 and x + 2 in F5 [x] modulo x2 . Are any of them primitive roots? [Hint: The orders divide 20.]