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Abstract Algebra Lecture Notes Abstract Algebra Lecture Notes David M. McClendon Department of Mathematics Ferris State University Fall 2018 edition 1 Contents Contents 2 0 Review of Math 324 4 0.1 Sets . 4 0.2 Relations . 12 0.3 Functions . 14 0.4 Cardinality . 19 0.5 Summary of proof techniques . 21 0.6 Outlines for common types of proofs . 23 1 Major problems of abstract algebra 33 1.1 Straightedge and compass constructions . 33 1.2 Polynomial equations . 44 1.3 Summary: the major questions we want to tackle . 56 2 Integers and rational numbers 57 2.1 N: the natural numbers . 57 2.2 Z: the integers . 64 2.3 Q: the rational numbers . 69 2.4 Divisibility . 73 2.5 The Euclidean algorithm and applications . 79 2.6 Congruence classes modulo n ....................... 86 3 Real and complex numbers 98 3.1 Theorems of Hippasus and Theatitus . 98 3.2 R: the real numbers . 101 3.3 C: complex numbers . 107 3.4 Fundamental Theorem of Algebra . 115 3.5 Complex roots, cubic equations and regular polygons . 117 2 Contents 4 Polynomial rings 126 4.1 Definition and basic properties . 126 4.2 Irreducibility tests . 135 5 Fields 141 5.1 Field extensions . 142 5.2 Algebraic extensions . 143 5.3 Linear algebra and field extensions . 146 5.4 Classical construction problems, revisited . 152 6 Morphisms 156 6.1 What is a homomorphism? . 156 6.2 Isomorphisms and invariants . 160 6.3 Ring homomorphisms . 165 6.4 Automorphisms . 173 7 Groups 174 7.1 An update on the big picture . 174 7.2 What is a group? . 177 7.3 Examples of groups . 187 7.4 Permutation groups . 195 7.5 Subgroups and cosets . 205 7.6 Normal subgroups and quotient groups . 209 8 Quintic equations 217 8.1 Solvability by radicals . 217 8.2 Galois groups . 220 8.3 Quintic equations, revisited . 227 Index 231 3 Chapter 0 Review of Math 324 Math 324 is an introduction to mathematical language: that means sets, relations and functions. It is also an introduction to proof. In the first four sections of this chapter, we review the language (and general theorems about that language you should know) that was developed in Math 324; in the last section, we discuss gen- eral methods to prove statements. Before we get started, here are some symbols and an abbreviation I use often: ∀ means “for all” ∃ means “there exists” ∧ means “and” ∨ means “or” ∼ means “not” s.t. is short for “such that” 0.1 Sets The fundamental objects of mathematics are called sets. A set is really just a list of objects (in math, the objects are usually numbers, or vectors, or functions). Definition 0.1 A set is a definable collection of objects. The objects which comprise a set are called the set’s elements. If x is an element of set E, we write x ∈ E; if x is not an element of set E, we write x∈ / E. 4 0.1. Sets Examples of sets (observe that sets are usually denoted by capital letters): A = {3, 5, 7, 9, 11} B = {1, 2, 3, 4, 5, 6} C = {3, 5, 7} The elements of set C described above are 3, 5 and 7. For the set A above, 3 ∈ A and 5 ∈ A but 8 ∈/ A. We often define a set without listing the elements (using English language). For example, the sets A, B and C given above could be described, respectively, by saying “let A be the set of odd numbers from 3 to 11”; “let B be the set of integers from 1 to 6”; “let C be the set of odd numbers from 3 to 7”. We also describe sets by using what is called set-builder notation: to describe the same sets A, B, C as above using set-builder notation, we would write (or say) A = {x : 3 ≤ x ≤ 11 and x is odd} B = {x : 1 ≤ x ≤ 6 and x is an integer} C = {x : 3 ≤ x ≤ 7 and x is odd}. The first statement above is interpreted as follows: it says that set A is equal to the set of numbers x such that (the colon means “such that” in mathematics) 3 ≤ x ≤ 11 and x is odd. Notice that this is exactly the set {3, 5, 7, 9, 11}. To show you a different kind of example: if you were defining some set of functions (instead of a set of numbers), then instead of x you’d write f, and then after the colon you’d describe what has to be true about f for the function f to be in the set. For example, the set D of functions whose derivative at x = 2 is positive could be described by writing D = {f : f 0(2) > 0}. For this set D, it would be valid to say that if g(x) = x3, then g ∈ D (because g0(2) = 3(22) = 12 > 0) but if h(x) = 3 − 4x, then h∈ / D (because h0(2) = −4 ≤ 0). Definition 0.2 The empty set, denoted ∅, is the set with no elements. 5 0.1. Sets Definition 0.3 Let E be a set. The power set of E, denoted 2E or P(E), is the set of all subsets of E. The elements of a power set are themselves sets. Example: if E = {1, 2}, then 2E = {∅, {1}, {2}, {1, 2}}. Venn diagrams A useful way to think about sets is to draw pictures called Venn diagrams. To draw a Venn diagram, represent each set you’re thinking about by a circle (or an oval, or a square, or a rectangle, or some other shape); think of an object as be- ing an element of the set if and only if it is inside the shape corresponding to the set. For example, a Venn diagram for the set A described above (recall that A = {3, 5, 7, 9, 11}) would be given by something like 2 4 6 8 10 12 1 3 5 7 9 11 A because the box describing A contains exactly the elements of A (nothing more and nothing less). Similarly, a Venn diagram representing the sets A, B and C from above at the same time would be something like 2 4 6 8 10 12 1 3 5 7 9 11 C A B 6 0.1. Sets Subsets and set equality Definition 0.4 Let E and F be two sets. • We say E is a subset of F , and write E ⊆ F , if ∀ x, x ∈ E ⇒ x ∈ F . If E is not a subset of F , we write E 6⊆ F . If E ⊆ F , we also write F ⊇ E and say that F is a superset of E. • We say E and F are equal, and write E = F , if E ⊆ F and F ⊆ E. If E and F are not equal, we write E 6= F . Example: {0, 1, 2} ⊆ {0, 1, 2, 4, 8} but {0, 1, 2} 6⊆ {0, 2, 4}. Note the difference between the symbols ∈ and ⊆: the first symbol should be preceded by an element, but the second symbol should be preceded by a set. Generally speaking, to say E ⊆ F means “everything in E also is in F ” or “E is inside F ”. If you draw a Venn diagram, to say S ⊆ T means that the shape corresponding to set S is completely inside the shape corresponding to set T . For example, for the sets A and C given in the preceding section, C ⊆ A since every element of C is also in A. To say two sets are equal means that they contain exactly the same elements. Example: {n ∈ N : n is an even prime} = {n ∈ N : n − 2 = 0} because the only element in each set is 2. Operations on sets Definition 0.5 Let E and F be sets. • The union of E and F , denoted E ∪ F , is defined as follows: E ∪ F = {x : x ∈ E or x ∈ F }. • The intersection of E and F , denoted E ∩ F , is defined as follows: E ∩ F = {x : x ∈ E and x ∈ F }. 7 0.1. Sets Definition 0.6 Let E and F be sets. • We say E and F are disjoint if E ∩ F = ∅. • The complement of E, denoted EC , is the set EC = {x : x∈ / E}. • The difference of E and F , denoted E − F and read “E minus F ”, is the set E − F = E ∩ F C . • The symmetric difference of E and F , denoted E 4 F , is the set E 4 F = (E − F ) ∪ (F − E). • The (Cartesian) product of E and F , denoted E × F and read “E cross F ”, is the set E × F = {(x, y): x ∈ E and y ∈ F }. The elements of a Cartesian product are called ordered pairs. We denote E ×E by E2. • Let E be a set and let n be a positive integer. The nth Cartesian power of E, denoted En, is the set of ordered n-tuples of elements from E: n E = {(x1, x2, ..., xn): ∀j, xj ∈ E}. Concepts: • ∪ is set language for “or”–the union of a bunch of sets is the set consisting of elements belonging to at least one of the sets. For example, using the sets described earlier, A ∪ B = {1, 2, 3, 4, 5, 6, 7, 9, 11} because all of the numbers listed there either are in A, or in B, or both. In terms of Venn diagrams, the union is usually thought of as a “MasterCard- symbol” shaped region that encompasses the sets whose union you are tak- ing.
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