Concrete Approach to Abstract Algebra by W. W. Sawyer

Author:W. W. Sawyer
Language: eng
Format: epub
Publisher: Courier Publishing
Published: 2018-08-14T16:00:00+00:00

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* If w = a + bJ, z = c – dJ, the identity, in modulus notation, states │w│ · │z│ = │wz│.

Chapter 6

Extending Fields

IN CHAPTER 5 we considered a way of extending a given field. The method used the idea of residue classes, but, as we saw in a lengthy discussion, it was often inconvenient to keep the final result in terms of residue classes. The residue classes gave us a way of showing that a certain type of calculating machine could be built, and indeed of building it. However, once having obtained the new machine, we shall often want simply to work it, and to forget what is inside it. So let us look at the fields we obtain by the procedure of chapter 5, and see just what this procedure does for us.

At the beginning of chapter 5, by considering residue classes modulo (x2 + 1), we obtained the symbols a + bJ where a and b are real numbers and J satisfies the equation J2 = – 1. a + bJ is commonly referred to as a "complex number." In order to work correctly with complex numbers, all you need to know is (i) that complex numbers obey the laws of algebra, (ii) that J2 = –1. Statement (i) here could be put, that complex numbers form a field. As the real numbers form a field, in passing from real numbers to complex numbers, we are not conscious of any change, so far as statement (i) goes. The main novelty lies in statement (ii), that J2 = – 1. So long as we are working with the real numbers, the equation x2 + 1 = 0 has no solution. The effect of passing to the complex numbers is to bring in a new symbol J, such that J2 + 1 = 0. We thus, so to speak, create a root for the equation x2 + 1 = 0. You will notice that the procedure of chapter 5 used residue classes modulo (x2 + 1).

In the same way, toward the end of chapter 5, to obtain a field in which x2 – 2 = 0 had a root, we considered residue classes modulo (x2 – 2).

Quite generally, if f(x) is an irreducible polynomial over a field F, we can obtain a field in which the equation f(x) = 0 has a root by considering residue classes modulo f(x).

(You should be able to prove this result, by observing the proofs on pages 98 and 105, for J2 + 1 = 0 and K2 – 2 = 0, and noting that the method used in these two particular proofs can be used in general. A proof will be given shortly, but it does no more than carry out the hint here given: if you can find the proof unaided, you will gain in insight and confidence.)

In speaking above of statements (i) and (ii) as embodying all you need to know to calculate with complex numbers, I should perhaps have added something you need to know is not so.

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