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 Wednesday, December 5 Main Idea: Things like the complex numbers. Key Words: Quaternions, Octonions, Sedenions. Goal: Learn about things bigger than the complex numbers. When we solved the quartic on Wednesday, I said since the solution was based on the chain S4 > A4 > K4 > {0,(12)(34} > 0 the process requires a quadratic, a cubic, and then two more quadratic. The solution of the cubic p(x) = (x - (ab+cd))(x-(ac+bd))(x-(ad+bc)) requires first a quadratic and then a cubic. Now the next quadratic gives you ab, cd, ac, bd, ad, bc Now the next quadratic gives you a, b, c, d, and some extraneous roots. The standard extensions of the rationals are now discussed. These all have a multiplicative norm. That is, a map n(x) into the positive reals such that n(xy) = n(x)n(y) Rationals c Reals c Complexes c Quaternions c Octonions c Sedenions. not not not not much ordered commutative associative at all The rationals are all numbers that can be expressed as a ratio of integers. Thus 3/4, 2/9, 0.333333..., 2 Pi/18 Pi, are all rational. Any repeated decimal is rational. So Express 38.123412341234 ..... as a ratio of integers. The reals are often thought of as all decimal numbers a1a2a3.b1b2b3b4..... but this is really not very satisfactory. If you take a number 0.99999999999999999... but not all nines and add it to a number 0.000000000000000... but not zeros, how to you write the answer. Does it start with 1.000000000... or does it start 0.999999999.... . To know how to start writing the number, we have to know what the numbers are maybe even out to several places beyond where we are transcribing the answer. One way the problem is approached is to say a number is something where there is a reason, or formula to tell what the nth place is. This gets cumbersome if you demand the decimal digits. Something like Pi is better expressed bu the rule that it is the ratio of the circumference to the diameter of a circle. This second approach is done by saying that the reals are the limit points of sequences of rationals. This makes addition and multiplication easy, since the (a1,a2,a3,...)+(b1,b3,b3...) = (a1+b2,a2+b2,a3+b3, ...) Since we can give the sum by giving something which converges to the sum. It is easy to show that if {ai} converges, and {bi} converges, then {ai+bi}, {ai*bi} also converge. The problem was that how can you say a sequence converges without giving the limit. This was done by Cauchy. He showed that a series converged it it satisfied the Cauchy condition. The other way the problem is handled was by Dedekind cuts. A dedekind cut is a partition of the rationals into two sets, a top half and a bottom half. Every rational is either in the top half, or bottom half. And every element is the top half is greater than every element in the bottom half. If you want to represent the square root of 2, the top half = all rationals whose square is greater than 2. The bottom half is all the rest. The complex numbers are all numbers of the form a+bi where i^2 = -1. The major thing about the complex numbers is that they are algebraically closed. Thus if p(x) = cnx^n + cn-1 x^n-1 + .... co is a polynomial with complex coefficients then p(x) = cn(x-r1)(x-r2) ... (x-rn). It factors completely. The major loss is that there is no order on the complexes. That is, there is no way to divide them into positive, negative, and zero so that the positive are closed under multiplication and addition. The quaternions are defined by quaternions = {a+bi+cj+dk} where i^2 = j^2 = k^2 = -1 and i clockwise is positive. counter clockwise is negative. k j The norm is defined by n(a+bi+cj+dk) = a^2+b^2+c^2+d^2. and the norm is multiplicative. n[x[{a,b,c,d},{p,q,r,s}]] 2 2 (d p - c q + b r + a s) + (c p + d q + a r - b s) + 2 2 (b p + a q - d r + c s) + (a p - b q - c r - d s) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = a p + b p + c p + d p + a q + b q + c q + d q + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 a r + b r + c r + d r + a s + b s + c s + d s 2 2 2 2 2 2 2 2 = (a + b + c + d ) (p + q + r + s ) = n(a,b,c,d) n(p,q,r,s). The conjugate of a+bi+cj+dk = a-bi-cj-dk and conjugation is multiplicative as well. _ Also x x = n(x). _ The quaternions have division. 1/x = x/n(x). The nicest thing about the quaternions is that they represent rotations. -1 If n(x) = 1, then x(0,a,b,c) x rotates the vector (a,b,c). The axis of rotation is the i,j,k coordinates of x. The amount of rotation is 2 ArcCos of the real part of x. The Octonions are not associative, not commutative, not ordered. They are best described as (a B) (e F) = ( ae+B.G, aF + h B - CXG ) (C d) (G h) ( eC+dG +BXF C.F + dh ) Where small letters are scalers and capital letters are vectors of length three. The dimension is 8. The origin of the Octonions is in Geometry. Cartesian geometry coordinates with the real numbers. If you do not have Desargueans property, they you cannot coordinatize with the reals, but they can be coordinatized with the Octonions. Finally, the Sedenions are degree 16. There is not much left. Assignment: 1. Find a dedekind which gives the cube root of 2. 2. Express 3.12121212.... as a rational number. 3. Find a quaternion which rotates through 30 degrees about the axis vector <1,2,3>.