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2 Introduction to Abstract Algebra 2 Introduction to Abstract Algebra We now give an axiomatic description of the fundamental objects of interest in this course. We’ll introduce the main ideas, and give examples of these algebraic structures. Definition 2.1. A group is a set G together with an operation , satisfying: ∗ [G1] a (b c) = (a b) c for all a,b,c G (* is assoc. wrt ), ∗ ∗ ∗ ∗ ∈ ∗ [G2] there is some e G st a e = e a = a for all a G (G has an identity element wrt ), ∈ ∗ ∗ ∈ ∗ [G3] given any a G there exists some b G such that a b = b a = e (every element of G has an inverse wrt ). ∈ ∈ ∗ ∗ ∗ Example 2.1. 1. Z, Q, R, C are groups wrt the usual addition 2. N is not a group wrt the usual addition (no identity) 3. Q 0 , R 0 , C 0 are groups wrt the usual multiplication, but Q, R, C are not (as 1/0 is not con- tained\{ } in\{ any} of\{ these} sets) 4. the set of even integers is a group wrt the usual addition 5. M(2, R), the set of all 2 2 real matrices is a group wrt matrix addition × 6. GL(2, R), the set of all real 2 2 invertible matrices is a group wrt matrix multiplication × 7. the set C = 1, i, 1, i is a group wrt the usual multiplication in C, where i = √ 1. 4 { − − } − If G is a group under and a b = b a for all a,b G (if is commutative on G) we say that G is an abelian group. ∗ ∗ ∗ ∈ ∗ Definition 2.2. A ring is a set R, together with a pair of operations, called addition (+) and multiplication ( ), st: · [AG] R is an abelian group wrt + (ie sat G1, G2, G3), [R1] multiplication is assoc in R,, [R2] the left and right distributive laws hold for all a,b,c R: ∈ a(b + c)= ab + ac, and (a + b)c = ac + bc. The additive identity of a ring R is denoted by 0, and is called the zero of R. If R has a multiplicative identity, we say that it is unital. The multiplicative identity of a unital ring R is denoted by 1, and is called its unity. We say that R is commutative if a b = b a for all a,b R. A zero divisor of a ring is an element a R st there exists some b R sat ab ·= 0 with· a and b both∈ nonzero. If R is commutative and has no zero∈ divisors we say that it is an∈ integral domain (ID). Definition 2.3. A field is a set F , together with a pair of operations, called addition and multiplication, sat [CR] F is a commutative ring, [F1] every non-zero element of R has a multiplicative inverse. We could say that a field F is a set that is an abelian group wrt addition and is an abelian group wrt and multiplication once we discard the zero. Alternatively, we could define it as an ID in which every nonzero element has an additive inverse. Example 2.2. 1. The familiar number systems Z, Q, R, C are all rings wrt the usual operations of addi- tion and multiplications. 2. Z is an ID wrt the usual operations of addition and multiplications. 3. Mn(Q) is a unital ring but is not an ID, being non-commutative and also having zero divisors. 4. Q(√3) = a + b√3 : a,b Q R is a field wrt the usual operations of multiplication and addition in R. { ∈ }⊂ 5 2.1 The Familiar Number Systems These are the usual numbers we’ve all seen before. The motivation for their construction is perhaps best viewed in the context of solving equations. We briefly describe the familiar number systems below. A complete description is beyond the scope of this course. 1. N = 1, 2, 3, 4, ... ...the natural numbers, { } 2. Z = ..., 3, 2, 1, 0, 1, 2, 3, ... ...the integers, { − − − } 3. Q = a/b : a,b Z,b =0 ...the rational numbers, { ∈ 6 } 4. R...the real numbers such as π,e, 3, 2, 0, 1/6, 1.571428358209504..., 0.3333333..., √11 etc. − 5. C = a + ib : a,b R,i = √ 1 ...the complex numbers. { ∈ − } These numbers satisfy the relation N Z Q R C. ⊂ ⊂ ⊂ ⊂ The integers are better behaved than the natural numbers in the following way. Suppose we’d like to solve the equation 2 + x = 1. Then this has a solution in Z, namely x = 1, but has no solution in N. While the usual addition is assoc in N, N has no identity element wrt this operation,− while Z does. In fact Z is a group wrt addition, so every equation of the form a + x = b can be solved for a unique x Z for given integers a and b. Z is formed by adding 0 (the additive identity) and the negative of every natural∈ number to N. Suppose now we wish to solve the equation 5x = 7. This equation has the solution x =7/5 in the rational numbers, but has no solution in Z. Z is an ID wrt the usual operations of addition and multiplication, but not every nonzero element of Z has a multiplicative inverse. If we add all these inverses to Z we form the rational numbers. In Q we can solve all equations of the form ax = b for integers a,b with a = 0. The rational numbers are a field. 6 This isn’t enough to be able to solve all equations, however. Consider the following equations: 1. x2 5=0, − 2. x3 + x + 1, 1 3. π−x . None of these can be solved in Q but they can all be solved in R. In fact there are many more real numbers that are not rational than those that are. For example, any decimal number representing a rational must either terminate (like 1.592) or become periodic (like 0.131313...). The essential difference between the reals and the rationals is that in R every non-empty subset S of R that has an upper bound in R has a least upper bound. For example the set S = x R : x2 < 3 has the least upper bound √3 in R, but not in Q. { ∈ } Even with all of these extra numbers added in, there still remain some equations that have no solution in R, for example 1. x2 +1=0, 2. x2 +5=0, 3. x4 + x3 + x2 + x + 1. These are all examples of polynomials. We’ll look at these objects in more detail later in the course. A solution to the first equation is given by x = √ 1. This solution is so distinguished that we give it a special symbol, namely i. In fact adding this number −i to the reals and then performing all possible computations with these generates the entire set of complex numbers. So solutions of the 2nd equation are given by x = i√5, and solutions of the 3rd equation by x = e2πki/5, k = 1, 2, 3, 4. The complex numbers are an example± of what is known as an algebraically closed field. This means that every polynomial with coefficients in R has a solution in C. This result is known as the fundamental theorem of algebra. 6 2.1.1 The Field of Fractions Q of the Integeral Domain Z The rationals can be constructed formally from the integers as follows. We identify the pair (a,b) with (c, d) in Z Z if ad = bd. Denote by a/b any pair (c, d) that satisfies this relation (so, for example, we don’t distinguish× between (1, 5) and (2, 10)). By convention we choose a/b to represent all elements that can be identified with (a,b) if a and b have no common factors other than 1. Let Q = a/b : a,b Z,b = 0 . Then Q has a ring structure with respect to the operations ± { ∈ 6 } a/b + c/d = (ad + bc)/bd and a/b c/d = ad/bd. · We leave it as an exercise to show that Q as defined is a ring wrt these operations. Note that the axioms of a ring can be shown to hold iff we assume already that Z is a ring. In fact Q is easily seen to be a field since if it has unity 1/1, is commutative and if a/b = 0 then a =0 so b/a is well defined and (a/b)−1 = b/a. Moreover, Z is contained in Q under the identification6 n n/16 for all n Z. Q is called the field of fractions of the integral domain Z. It is the unique smallest field7→ that contains Z∈. This construction can be applied to any ID R. We usually denote the corresponding field of fractions by FR. 2.1.2 Construction of the Complex Numbers from the Real Numbers The complex numbers can be constructed from the reals in a similar way. Consider all pairs (a,b) R R and denote the set of all such pairs by C. Then C forms a ring wrt the following operations of addition∈ × and multiplication (where any arithmetic that occurs in each component is computed in R): (a,b) + (c, d) = (a + b,c + d) and (a,b) (c, d) = (ac ad,ac + bd). · − Again we leave it as an exercise to show that the axioms of a ring hold, noting that the proof depends of the fact that R is itself a ring.
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