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Mathematics Department Stanford University

Math 61CM/DM { Basic algebraic structures

The purpose of this handout is to provide some basic denitions and a brief discussion of some key ideas and objects in algebra. One main new idea you will be studying in class is the notion of aeld. This is a concept which generalizes the algebraic properties of the real numbers (as well as the rational numbers and the complex numbers). However, mathematicians nd it useful to abstract the properties of these familiar elds into a general denition. It turns out that there are a lot of other interesting examples; we present a few of these below. One of the points of this is that a substantial portion of linear algebra can be done in the setting of general elds, rather than how it is more customarily done, over the real numbers. This will be especially pertinent to the students in 61DM. For the 61CM students, you should view this as a good way to get accustomed to a slightly dierent level of abstraction, and you should also use this as an opportunity to get used to the fact that the basic results and methods in linear algebra are really \algebraic", i.e., only depend on the handful of axioms in the denition of a eld. In mathematics, the most fundamental objects aresets. These are just collections of objects. We will be concerned with either nite or certain innite sets. (It turns out that there is quite a lot that can be said about sets in general, but that's another story.) However, sets are kind of static objects { they just sit there. One can dene progressively more sophisticated objects by prescribing certain sets of rules, or axioms, about how the elements of the sets are related to or interact with one another. We are going to describe a few dierent types of rules, each slightly more specic than the last, leading to dierent types of algebraic objects: semigroups, groups and then elds. This handout collects these denitions, discusses a few examples and basic results. Denition 1A semigroup with unit(G;;e)is a set of objectsGcontaining a distinguished elemente2G, along with a map:GG!G, with the following properties:

1. (Associativity) For allx;y;z2G,x(yz) = (xy)z.

2. (Units) There exists an elemente2Gsuch that for allx2G,xe=x=ex.

You should think of the operationas being `multiplication'; it is a way of combining elements of the setGto get other elements. Here is a basic example: letG=f1;2;3;:::g, andordinary multiplication. The `identity element'eis simply 1. It is obvious that the two rules hold. Here is another example: now letG=f0;1;2;:::g, but this time, suppose thatis the symbol which corresponds to addition. Nowe= 0 and once again the two rules hold. This is one of the most primitive algebraic structures: we have a set and a way of combining elements in it. We only have one \extra", which is the existence of a unite. (One can also talk about semigroups without units; there are even some interesting examples!) Lemma 1In any semigroup with(G;;e), the uniteis unique. Proof:Supposee;f2Gare two possibly dierent elements which both satisfy the properties of a unit. Thene=efsincefis a unit, andef=fsinceeis a unit. Combining these, we see thate=f.This means that one can talk abouttheunit, i.e. given (1) and (2),e is unique. It is a bit more interesting if one adds one further property: Denition 2A group(G;)is a semigroup with unit(G;;e)with one additional property:

1. (Inverses) For everyx2G, there exists an elementy2Gsuch thatxy=e=yx.

Similarly to the lemma above, inverses are also unique. Lemma 2Suppose that(G;)is a group. Ifx2Gthere exist elementsy;z2Gsuch that yx=e=xz. Theny=z.

Try to prove this yourself!

It is usually simpler to just say thatGis a group, without explicitly mentioningande. Groups turn out to be really fundamental and extraordinarily important objects in mathemat- ics; they are the basic way to describesymmetry. Groups are widely used in physics, chemistry and elsewhere.

Here are a few interesting groups:

1. (R;+), the set of all real numbers with addition;

2. (Z;+), the set of all integers with addition;

3. (Q;+), the set of all rational numbers with addition;

4. (Q[p2];+), the set of all numbers of the forma+bp2 wherea;b2Q;

5. (Rn;+), the set of all vectors withncomponents under addition;

6. (R+;), the set of all positive real numbers with multiplication;

7. (Z=(nZ);+), the set of integersmodulon, wherenis an integer greater than 2, under

addition;

8. The group of all rotations around the origin inR2;

9. The group of all rotations around the origin inR3.

For 7) here, we recall that as a set,Z=(nZ) is identied withf0;1;:::;n1g(remainders when dividing byn), and + means ordinary addition inZ, reduced modulon, so e.g. in (Z=(5Z);+),

2 + 4 = 1. It is maybe less confusing to write this set asf[0];:::;[n1]gso we don't think of

its elements as ordinary integers. Then [2] + [4] = [1], etc. All but the last of these examples of groups have an important extra property: Denition 3A commutative, or abelian, groupGis one in whichxy=yxfor allx;y2G. Noncommutative groups (and semigroups) do play a role, even in this class. The group of rotations inR3is a really important example; another example of a noncommutative semigroup is the setMnofnnmatrices, wheredenotes matrix multiplication. This is noncommutative whenn2. Another example we shall see again is the set of all permutations of a nite setS. Permutations form a group: if we permuteSthen permute again, then the composition of the two permutations is another permutation. The order in which we compose is important, and we would get a dierent answer ifShas at least three elements.

Finally we come to the denition of a eld:

Denition 4A eld(F;+;)is a setFwith two distinct maps+ :FF!Fand:FF!

F, which satisfy

1.(F;+)is a commutative group, with unit0.

2.(F;)is a commutative semigroup with unit1.

3.16= 0and ifx2Fandx6= 0, thenxhas a multiplicative inverse (i.e., an elementy

such thatxy= 1 =yx).

4. The distributive law holds:

x(y+z) =xy+xz: One usually writesxfor the additive inverse (inverse with respect to +), andx1or 1=xfor the multiplicative inverse. Examples then include (R;+;), (Q;+;), as well as complex numbers (C;+;). It turns out that (Z=(nZ);+;) is a eld if and only ifnis a prime number! Furthermore,

Q[p2] =fa+bp2 :a;b2Qg

is also a eld! It is obvious that this set is closed under addition, and with just one line of computation, you can see that it is also closed under multiplication. Furthermore, it obviously contains all additive inverses. To see that it contains all multiplicative inverses, suppose that a+bp26= 0, and observe that computing inR, (a+bp2)

1=abp2

a

22b2= (a22b2)1a(a22b2)1bp2:

The point here is that (a22b2)1aand(a22b2)1bare both rational, anda22b26= 0. Here is an example of a general result about elds:

Lemma 3If(F;+;)is a eld, then0x= 0for allx2F.

Proof:Since 0 = 0 + 0, we have

0x= (0 + 0)x= 0x+ 0x;

so

0 =(0x) + (0x) =(0x) + (0x+ 0x) = ((0x) + 0x) + 0x= 0 + 0x= 0x;

as desired. On the last line, the rst equation is that(0x) is the additive inverse of 0x, the second substitutes in the previous line, the third is associativity, the fourth is again that (0x) is the additive inverse of 0x, while the fth is that 0 is the additive unit. Notice that this proof uses the distributive law crucially: this is what links addition (0 is the additive unit!) to multiplication.

For more examples, see Appendix A, Problem 1.1.

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