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by: Cassidy Grimes


Cassidy Grimes

GPA 3.51


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Class Notes
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This 4 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 700 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/229540/math-700-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15
CLASS NOTES MATHEMATICS 700 FALL 1999 RALPH HOWARD 1 FIELDS 11 The De nition of a Field The basic objects of study in linear algebra are vector spaces over elds and linear maps between vector spaces and one of our rst prices of business is to de ne all of these terms If R is the set of real numbers then the basic example of a vector space is the set R of n tuples 1 xn where each is an element of R Actually we will usually write elements of R as column tuples 1 xn as this is more consistent with doing matrix algebra I am assuming all of you have seen some linear algebra at least in the form of doing matrix computations where the elements of the matrices are real or complex numbers The matrix theory part of the class will look very much like what you have seen but a large part of what we do will not evolve matrices with the di erence that the elements of the matrices are not restricted to being real or complex numbers but can come from more general sets called elds The basic idea is that a eld F is a set with two operations addition denoted by and multiplication denoted by or just concatenation ie a b ab that satisfy all the rules of high school algebra More precisely De nition 11 A eld F is a set F with two binary operations and so that 1 The operations and are both commutative and associative zy yz zyz zyx xy yz 2 Multiplication distributes over addition zy z xy x2 3 There is a unique1 element 0 E F so that for all z E F x 0 0 z x This element will be called the zero of F 1It is not hard to show the assumption of the uniqueness can be dropped For if 0 and 0 are elements so that 10 z and 10 I then 0 00 0 0 0 which implies uniqueness l rb There is a unique2 element 1 E F so that for all z E F x 1 1 w x This element is called the identity of F 0 31 1 This implies F has at least two elements For any x E F there is a unique3 em 6 F so that z em 0 This element is called the negative of x And from now on we write x 7y as x 7 lfO 31 z E F there is a unique4 element z 1 E F so that Q 7 zmil xilm 1 We will also denote 4 by 1z and ya 1 by The element z 1 is called the inverse of We will usually just refer to the eld F77 rather than the eld F For any eld F we can View the positive integer n as an element of F by setting n111 W n terms Then for negative 71 we can set 71 7771 where in is de ned by the last equation That is 74 71 1 1 1 In most respects all of the basic algebra you know works as usual in a eld As a sample of this Proposition 12 Let F be a eld Then for all a b E F 1 a 0 0 2ab0ifandonlyifaOorb0 3 x2a2 impliesmaorz7a 41fad7bc 31 0 then aby 6 im lies ziwi afice czdy f p Tadibc yiadibc Proof An exercise you should do if you are not familiar with this circle of ideas D 12 Examples of Fields 2Again the assumption of uniqueness can be dropped 3Again the assumption of uniqueness can be dropped 1fz y 0 and 12 0 thenyy0yzz zyz0zzi 4And yet again the assumption of uniqueness can be dropped 3 121 The rational numbers This is the set of ratios ab of integers a b with b 31 0 Note that integers can be either positive or negative This is the most natural example of a eld 122 The real numbers For lack of a shorter de nition this is the collection of all decimal numbers R I am assuming that you know the basic properties of the real numbers or are at least learning about them in Math 703 One advantage of the real numbers over the rational numbers is that every positive real number has a real numbers as a square root while not every rational number has a rational number as a square root 123 The eomplew numbers Let t x71 so that t2 71 Then the complex numbers C is the set of all numbers z ty where z and y are real numbers and addition and multiplication are done in the natural way That is 1 91 2 92 1 x2 91 92 1 91 92 92 9192 i 9192 192 291 The rule for multiplication is what is obtained by expanding the prod uct 1 yltx2 ygt using that t2 71 and grouping the terms with an 239 together What is not quite obvious if you have not seen it before is that if z ty is a nonzero complex number that it has an in verse To get the inverse of z ty we use the trick of rationalizing the denominator77 by multiplying by z 7 ty call the complex conjugate of z ty That is 1 z 7 ty z 7 ty z 7y z y T y y T 962 T z292 sz 39 Note that 2ty 31 0 means that z 31 0 or y 31 0 so that 22y2 gt 0 and therefore the above calculation works for any nonzero complex number The importance of the complex numbers is that not only does every complex number 2 have a square root that is also a complex number but if pz anz an1z 1 a0 is a polynomial with complex coef cients and an 31 0 then the equation pz 0 will always have at least one complex solution This fact is referred to as the fundamental theorem of algebm 13 Quadratic number elds This is not anything that we will use other than in passing but it is interesting to see examples that are not as familiar as the ones above Let n be an integer either positive or negative that does not have a rational square root That is 2 n does not have a solution with z ab a rational number This implies


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