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1.A Fields

The reader is expected to be familiar with several elementary properties of the real numbers and complex numbers, such as associativity, commutativity, and existence of inverses, all of which are listed below. If $F$ is the set of either the real numbers ${\mathbb{R}}$ or the complex numbers ${\mathbb{C}}$ , then they each have an addition $+$ and a multiplication $\cdot$ , and we will use the following facts without proof:

F1.

Addition is a binary operation: if $x,y\in F$ then $x+y\in F$ .
F2.

Multiplication is a binary operation: if $x,y\in F$ then $x\cdot y\in F$ .
F3.

Addition is commutative: if $x,y\in F$ then $x+y=y+x$ .
F4.

Multiplication is commutative: if $x,y\in F$ then $x\cdot y=y\cdot x$ .
F5.

Addition is associative: if $x,y,z\in F$ then $(x+y)+z=x+(y+z)$ .
F6.

Multiplication is associative: if $x,y,z\in F$ then $(x\cdot y)\cdot z=x\cdot(y\cdot z)$ .
F7.

There is an additive identity in $F$ : $\exists 0\in F$ such that $\forall x\in F$ we have $x+0=x$ .
F8.

There is a multiplicative identity in $F$ , distinct from the additive identity: $\exists 1\in F$ such that for any $x\in F$ we have $x\cdot 1=x$ .
F9.

There exists additive inverses in $F$ : if $x\in F$ then there exists a $y\in F$ such that $x+y=0$ .
F10.

There exists multiplicative inverses in $F$ for every element other than 0: if $0\neq x\in F$ then there exists a $y\in F$ such that $x\cdot y=1$ .
F11.

Multiplication distributes over addition: if $x,y,z\in F$ then $x\cdot(y+z)=x\cdot y+x\cdot z$ .

If $F$ is any set which has an addition $+$ and multiplication $\cdot$ obeying these rules, then the triple $(F,+,\cdot)$ is called a field; often we simply call $F$ a field, with the understanding that $+$ , $\cdot$ are vital parts of the definition. These rules are called the field axioms. While you are not expected to memorize these axioms for an exam, but you should know the meaning of the emphasized words, and you should be able to check whether a given axiom holds in a given situation; see the examples and exercises below.

Roughly speaking, a field is a set of “numbers” in which we can, in some sense, add, subtract, multiply and divide, and in which all of the usual laws of arithmetic are satisfied. For most of this module, whenever you read the word “field”, you will usually have no problem if you simply think of either the real numbers $F={\mathbb{R}}$ or the complex numbers $F={\mathbb{C}}$ .

Note that we sometimes write $x y$ instead of $x\cdot y$ .

Example 1.1.

i.

The set of rational numbers $\mathbb{Q}=\{\frac{a}{b}\;|\;a,b\in\mathbb{Z},b\neq 0\}$ with the usual addition and multiplication is a field. All of the axioms are taught in most schools at an early age. For example, verification that addition is a binary operation amounts to checking: If $a,b,c,d\in\mathbb{Z}$ and $b,d\neq 0$ then $\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$ . And since $a, b, c, d$ are integers, so are $ad+bc$ and $b d$ . Finally, since $b,d\neq 0$ , we know $bd\neq 0$ . Therefore we have justified that if $x,y\in\mathbb{Q}$ then $x+y\in{\mathbb{Q}}$ . In particular, F1 is satisfied.
ii.

The set of integers $\mathbb{Z}=\{\cdots,-2,-1,0,1,2,3,\cdots\}$ , with the usual multiplication and addition, is not a field. All axioms are satisfied except for F10. To prove that F10 is not satisfied, we need to find a counter-example. Let’s try $x:=2$ . Then $x\in\mathbb{Z}$ , and for any $y\in\mathbb{Z}$ , we have that $x\cdot y$ is an even integer. But that means $x\cdot y$ is not equal to 1 for any $y\in\mathbb{Z}$ . So we conclude that $2$ does not have a multiplicative inverse (in $\mathbb{Z}$ ). And since $2\neq 0$ , this proves that F10 is not satisfied for $F=\mathbb{Z}$ .
iii.

The set of two elements $\{0,1\}$ , where addition and multiplication are taken “modulo 2”, is a field. For example, $1+1=0$ , and $1\cdot 0=0$ , etc. One can verify that this satisfies the above axioms, and is therefore another example of a field; it is called the field with two elements, and is often denoted either $\mathbb{Z}/2\mathbb{Z}$ or $\mathbb{F}_{2}$ .

Exercise 1.2:

Another example of a field is the set of conjugacy classes of integers modulo 5, namely $\mathbb{F}_{5}:=\{0,1,2,3,4\}$ . For example, $3+4=2$ and $2\cdot 4=3$ . For each non-zero element of this field, find its multiplicative inverse.

[End of Exercise]

As you read these axioms, you might think that their abstract nature is a negative thing. But try to recall your first experience with mathematics: when you were learning about numbers. You may have been shown many sets of three objects - three balls, three dogs, three pencils - and, gradually, you learned to recognize the property that they had in common, namely, their “threeness”. For most of your life you have been comfortable with the abstract concept that we call the number 3. Here the procedure is similar: we are taking several familiar situations (in this case, ${\mathbb{Q}}$ , ${\mathbb{R}}$ , and ${\mathbb{C}}$ ), and we are recognizing some things that they all have in common (in this case, these axioms), and then naming the abstract concept (in this case, we use the word “field”). This is the process of abstraction, and as you’ve already discovered with the number 3, it can be very helpful!