Exercises 2

1. Show that the set of all subsets of a set S forms a ring under the operations A + B = A ∪ B - A ∩ B and A . B = A ∩ B for any subsets A, B ∈ S. This is called a Boolean ring.
   [Use Venn diagrams to check the ring-theory axioms.]
   Does this ring have an identity element? Which elements of the ring have multiplicative inverses?
   If instead we define addition by A + B = A ∪ B do we then get a ring ?

   Solution to question 1

2. By considering the product of "pure quaternions" (of the form bi + cj + dk) show how the scalar product and the vector product of vectors in R³ can be obtained from quaternionic multiplication.

   Show that the set R³ under vector addition and the vector product × is not a ring.

   Solution to question 2

3. Prove that the set of real polynomials a₀ + a₁x + a₂x² + ... + a_nxⁿ for which a₀ = a₁ = 0 is a subring of R[x]. Is it an ideal?

   Prove that the set of all real polynomials a₀ + a₁x + a₂x² + ... + a_nxⁿ for which the sum a₀ + a₁ + a₂ + ... + a_n = 0 is an ideal of R[x].
   [Hint: such a polynomial satisfies p(1) = 0.]

   Solution to question 3

4. Prove that the set { r + s√2 | r, s ∈ Q} is a field under real addition and multiplication. Prove that it is the smallest subfield of R which contains √2.

   Solution to question 4

5. Let R be the ring of elements of the form {a + bx | a, b ∈ Z₂} with arithmetic modulo 2 and multiplication using the "rule" x² = 1. Prove that this is not a field.

   Let R be the ring of elements of the form {a + bx | a, b ∈ Z₃} with arithmetic modulo 3 and multiplication using the "rule" x² = x + 1. Prove that this is a field with 9 elements.

   Solution to question 5

6. Show that in any finite field the additive order of any non-zero element must be a prime.
   Prove that in any finite field the additive orders of any two non-zero elements are the same. (This is called the characteristic of the field.)

   Solution to question 6

7. Let p be a prime number and k ∈ N. Consider the set R = {a₀ + a₁x + ... + a_k-1x^k-1 | a₁ , a₂ , ... , a_k ∈ Z_p} with arithmetic modulo p and multiplication using the "rule" x^k = q(x) for some fixed polynomial q(x) ∈ Z_p[x]) of degree < k.
   Prove that R is a ring with p^k elements.
   If the polynomial p(x) = x^k - q(x) can be factorised in Z_p[x] as a product of two lower degree polynomials r(x) and s(x), prove that R has zero-divisors.

   Prove that the polynomial p(x) = x² - x - 1 of Question 5 cannot be written as a product of linear factors in Z₃[x].

   Solution to question 7