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 = AB - AB and A . B = AB for any subsets A, BS. 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 = AB 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 R3 can be obtained from quaternionic multiplication.

Show that the set R3 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 R[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 R[x].
[Hint: such a polynomial satisfies p(1) = 0.]

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

5. Let R be the ring of elements of the form {a + bx | a, bZ2} 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, bZ3} 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 kN. Consider the set R = {a0 + a1x + ... + ak-1xk-1 | a1 , a2 , ... , akZp} with arithmetic modulo p and multiplication using the "rule" xk = q(x) for some fixed polynomial q(x) ∈ Zp[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 Zp[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].

SOLUTIONS TO WHOLE SET
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JOC/EFR 2004