Math 203 Extra Stuff, 1/21/05

Math 203 – Extra stuff, 1/21/05

About Fields

You don’t have to know what fields are for linear algebra, but it helps. You’ll need it if you do more mathematics. This definition is standard.

What’s a Field?

A field is any set K of “numbers” with two operations, called + and ∙ , with all of these properties:

(A) To every pair x and y of numbers (that is, elements of K) there is associated a number x+y, called their sum, in such a way that

(1) K is closed with respect to + : If x and y are in K, then x + y is in K.

(2) Addition is commutative: For every x and y in K, x + y = y + x.

(3) Addition is associative: For every x, y, and z in K, x + (y + z) = (x + y) + z.

(4) There exists a unique number called 0 (“zero”) in K such that for every x in K, x+0=x and 0 + x = x.

(5) For every x in K, there is a unique number –x in K (called “minus x”) such that x+(–x)=0 and (–x) + x = 0.

(B) To every pair x and y of numbers there is associated a number x ∙ y , called their product and often written xy, in such a way that

(1) K is closed with respect to ∙ : If x and y are in K, then x ∙ y is in K.

(2) Multiplication is commutative: For every x and y in K, x ∙ y = y ∙ x.

(3) Multiplication is associative: For every x, y, and z in K, x ∙ (y ∙ z) = (x ∙ y) ∙ z.

(4) There exists a unique number called 1 (“one”) in K, different from 0, such that for everyx in K, x ∙ 1 = x and 1 ∙ x = x.

(5) For every x in K except x = 0, there is a unique number called x–1 in K (called “x inverse,” also written 1/x) such that x ∙ (x–1) = 1 and (x–1) ∙ x = 1.

These requirements are called the “field axioms.” They are basically the rules of algebra we have been using for years.

The First Two Examples

1. The set of real numbers, R, is a field if we use the usual operations, addition and multiplication. The special words in the axioms (zero, one, minus x, x inverse) all have the meanings we are used to.

We often talk about “the field R”, but it’s important to remember that the choice of operations is part of the definition of the field. If we were pedantic we would talk about “the field R with the usual +, ∙.” If you made up a couple of silly operations to use in place of addition and multiplication, then R probably wouldn’t be a field with your operations. But people don’t do that much, so if we just refer to “the field R” nobody is really confused.

2. The set of complex numbers, C, is also a field with the usual addition and multiplication.

Why do we define fields?

The (first) main reason for defining a field is to save time. Suppose we work out some theory about, say, real numbers. (That’s what we’re doing in linear algebra this week.)

But let’s say that we’re careful to use only those facts about real numbers that are included in the field axioms. If we know other special facts about real numbers, we don’t use them. (Ihave been careful in linear algebra only to use the field axioms, although I haven’t made a point of it.)

Then we know that everything we do will work for complex numbers, too. We only used the field axioms, and they’re true for C too, so everything still works. And, if we think of other fields, our theory will work for them, too.

In linear algebra we are just referring to “numbers.” Now we know that “numbers” means elements of some field that we haven’t specified. The theory works if the numbers are drawn from any field.

More Examples of Fields

3. The rational numbers, Q. Rational numbers are numbers of the form a/b where a and b are integers (and b isn’t zero). For example, 2/3 is a rational number, and so is –22/7, but p isn’t a rational number and neither is .

It turns out that the rational numbers are the ones whose decimal expansions either terminate, or repeat. The expansions of p and don’t terminate or repeat, so they aren’t rational. The important things are (1) the sum of any two rational numbers is rational, (2) the product of any two rational numbers are rational, (3) zero and one are rational, (4) if x is rational then so is –x, and (5) if x is rational so is 1/x.

4. The integers mod p, ZP, if p happens to be prime. (If this doesn’t mean anything to you, don’t worry. But it is an example that shows that unusual fields can occur, and that fields can be finite.)

Examples of Non-fields

1. The integers, Z, are NOT a field if you use the usual definitions of addition and multiplication. The problem is axiom B(5), about multiplicative inverses. The integers do have inverses (the inverse of 3 is 1/3) but they aren’t in Z, so Z isn’t a field.

2. The integers mod n, Zn, if n isn’t prime (again, because in that case some numbers in Zn don’t have multiplicative inverses in Zn).

3. The quaternions (whatever they are) (because multiplication isn’t commutative).

Quick summary: A field is an environment in which a four-function calculator ( + – ´ / ) works the way you expect it to.

Some Problems

1. Show that if x, y, and z are elements of any field, then these things are true:

a. If x + y = x + z, then y = z.

b. (–1) ∙ x = (–x) (Show that (–1) ∙ x has the property that –x is supposed to have; but –x is supposed to be the unique number with that property)

c. (–x) ∙ (–y) = xy

d. If xy = 0, then either x = 0 or y = 0 (or both)

(These are all usual rules of algebra, but they aren’t among the field axioms. So, it is nice to know that they follow from the field axioms, so that they are still true in every field. Some of them are harder to prove than others.)

2. (if you know what Zn means) Show that if n isn’t prime, then some elements of Zn don’t have multiplicative inverses. So, Zn can’t be a field.

3. (Much harder) Show that if n isn’t prime, then Zn can’t even be contained in a field. (Hint: Use problem 1d. That rule holds for x and y in any field. Does it always hold when x and y are in Zn ? )

4. (Not as hard, but maybe tedious. This gets used in number theory.) Let K be the set of real numbers of the form , where a and b are both rational numbers, is a field. (Just check all the field axioms. B1 requires attention, but the tricky one is B5.)

5. (Very, very hard) Is there a set K that satisfies all of the axioms of a field except A2? (To solve this problem, you must do one of two things: give an example of a field-like thing in which addition isn’t commutative but everything else works, OR, give a proof of A2 using only the other field axioms.)

Why do we study fields, really?

To understand roots of equations. If we study the fields that are contained in R but are bigger than Q — like the one in problem 4 — we can learn a lot about which polynomials have roots and where to look for the roots. But that’s not for linear algebra.

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