Rings and Fields

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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 ?

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 3 can be obtained from quaternionic multiplication.

Show that the set 3 under vector addition and the vector product is not a ring.

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

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

4. Prove that the set { r + s2 | r, s } is a field under real addition and multiplication. Prove that it is the smallest subfield of which contains 2.

5. Let R be the ring of elements of the form {a + bx | a, b 2} 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 3} with arithmetic modulo 3 and multiplication using the "rule" x2 = x + 1. Prove that this is a field with 9 elements.

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.)

7. Let p be a prime number and k . Consider the set R = {a0 + a1x + ... + ak-1xk-1 | a1 , a2 , ... , ak p} with arithmetic modulo p and multiplication using the "rule" xk = q(x) for some fixed polynomial q(x) p[x]) of degree < k.
Prove that R is a ring with pk elements.
If the polynomial p(x) = xk - q(x) can be factorised in 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 3[x].

SOLUTIONS TO WHOLE SET
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JOC/EFR September 2004