301 Abstract Algebra 2:00 - 2:50 MWF Hentzel 432 Carver hentzel@iastate.edu Web Page http://www.math.iastate.edu/hentzel/class.301 Text: A First Course in Abstract Algebra, sixth Edition by John B. Fraleigh Course: Chapter 1,2,3,4,5 Monday, October 22. Main idea: You have not even looked at my proof! How can you say that it is wrong? Key words: Constructible, point, line, Quadratic extension. Rational Plane, Goal: Explain the rules of Geometric Construction and examine quadratic extensions. The rules of Geometric Construction are: (1) If you have two points, you can construct the line through those two points. (2) If you have a point P and a distance r, you can draw the circle of radius r about the point P. (3) When two lines intersect, you can identity the point of intersection. (4) When two circles intersect, or a line and a circle intersect, you can identity the point of intersection. You have two start somewhere. The bare minimum would be two points. The distance between them is called 1. You could just as well call it Pi. But then every distance would simply be multiplied by Pi. One can construct the line through the points and mark off the unit distances on the line. -------|-------|-------|-------|-------|-------|-------|------- -3 -2 -1 0 1 2 3 One can construct a perpendicular at 0 and create the y-axis also. | - 2 | | | - 1 | | | -------|-------|-------|-------|-------|-------|-------|------- -3 -2 -1 | 1 2 3 | - -1 | | | - -2 | | | And then one can construct more perpendiculars, both horizontals and verticals to get the integer grid. | | | | | | | | | | | | | | | | | | | | | -------|-------|-------|-------|-------|-------|-------|------- | | | (0,2) | | | | | | | | | | | | | | | | | -------|-------|-------|-------|-------|-------|-------|------- | | | (0,1) | | | | | | | | | | | | | | | | | -------|-------|-------|-------| ------|-------|-------|------- (-3,0) (-2,0) (-1,0) (0,0) (1,0) (2,0) (3,0) | | | | | | | | | | | | | | -------|-------|-------|-------|-------|-------|-------|------- | | | (0,-1) | | | | | | | | | | | | | | | | | -------|-------|-------|-------|-------|-------|-------|------- | | | (0,-2) | | | | | | | | | | | | | | | | | Now it is possible to get all the rational points on the x axis by. (1,s) /| / | / |s / | r/(r+s) / | --------------------/--------------------- | / s/(r+s) | / r| / | / | / |/ (0,-r) And so we can construct all the points (x,y) with rational coordinates. We call these points the "rational plane" The rational numbers are indicated by Q. The theory of construction is this. The geometric construction is a sequence of operations. These operations are of the following four types. (1) draw a line (2) draw a circle (3) locate the intersection of two lines (4) locate the intersection of a line and a circle or two circles. After doing enough of these operations, (a finite number) one claims he has succeeded in constructing the point, line, or whatever that was needed. At the start, the only known points, lines, and circles can be expressed using rational numbers. (a) the only known points are the points with both coordinates rational. (* rational points *) (b) the only known lines are of the form ax+by=c with a,b,c all rational. (* rational lines *) (c) the only known circles are of the form (x-a)^2 + (y-b)^2 = r^2 with a,b,r^2 rational. (* rational circles *) Now we want to say a few words about fields. The familiar fields are The Rationals c The reals c Complexes. For our purposes here, a field is a subset of the reals which is closed under addition, multiplication, subtraction, and division. There are lots and lots of fields existing between the Rationals and the reals. Example: Q[Sqrt[3]) = { a+bSqrt[2] | a,b are rational} (add) a+bSqrt[2 + c+dSqrt[2] = (a+c)+(b+d)Sqrt[2]; (subtract) a+bSqrt[2 - c+dSqrt[2] = (a-c)+(b-d)Sqrt[2]; (multiply) (a+bSqrt[2])*(c+dSqrt[2]) = (ac+2bd)+(ad+bc)Sqrt[2]; a+bSqrt[2] a+bSqrt[2] c-dSqrt[2] (ac-2bd)+(-ad+bc)Sqrt[2] (divide) ------------ = ---------- * ---------- = ----------------------- c+dSqrt[2] c+dSqrt[2] c-dSqrt[2] c^2-2d^2 ac-2bd -ad+bc = --------- + --------- Sqrt[2] c^2-2d^2 c^2-2d^2 From the above you can see the same process works to show that Q[Sqrt[3]], Q[Sqrt[5]], Q[Sqrt[7]], etc all give fields. Note also and the same argument does not really depend on Q at all. Suppose F is closed under addition, subtraction, multiplication, and division. And suppose that t is some element of F which does not have a square root in F. F[Sqrt[t]] = { a+bSqrt[t] | a,b are in F} is a field bigger than F. that contains a square root of t. One can imagine repeating this process over and over to get increasingly larger fields. Q = Fo c F1 c F2 c F3 ..... c Fi c Fi+1 .... Where Fi+1 = Fi[Sqrt[ti]] where ti is in Fi. Now, given any one of these fields Fi we can talk about Fi points, Fi lines, and Fi circles. During a construction, we start with rational points, lines, and circles. The intersections can generate non rational points. But any such point will be in Q[ Sqrt[t] ] for some rational t. In general, at any time we will be dealing with Fi points, lines, and circles. Any point we locate will be a point in Fi[ Sqrt[t] ] for some t in Fi. Furthermore, and line or circle generated from that point will be a Fi[ Sqrt[t] ] line or circle. And continuing this way, the construction maps out a sequence of fields Q = F0 c F1 c F2 ... c Fi c Fi+1 c ... where each Field is a quadratic extension of the previous. As complicated as elements in these fields appear to be, they have a universal property which we will show. That property is that they are always the root of a polynomial over the rationals, and the degree of the minimal such polynomial is a power of 2. This means that every point (x,y) we can construct will have to have an x and a y whose minimal polynomials over the rationals will have degree 2, 4, 8, 16, etc. We cannot construct (1, CubeRoot[3]) for example. The minimal polynomial satisfied by CubeRoot[3] is x^3-3 = 0. This is of degree 3, not a power of 2. Any construction which would lead us to the point (1,CubeRoot[3]) is impossible. Assignment. (1) Find the inverse of 3+5 Sqrt[2] in Q[Sqrt[2]] (2) Find the minimal polynomial of Sqrt[2]+Sqrt[3] in (Q[ Sqrt[2] ])[ Sqrt[3] ] = Q[ Sqrt[2], Sqrt[3] ]. (3) Find the inverse of Sqrt[2]+Sqrt[3] in Q[ Sqrt[2], Sqrt[3] ]