Description Details Hashtags Report an issue Book Description This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields.
The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups.
An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions.
Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over exercises, half of which have detailed solutions provided. Mathematics instructors, algebraists, and historians of science will find the work a valuable reference.
Download A Book Of Abstract Algebra books , Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. Download Basic Algebraic Systems books ,. Download Basic Abstract Algebra books , Thought-provoking and accessible in approach, this updated and expanded second edition of the Basic Abstract Algebra provides a user-friendly introduction to the subject, Taking a clear structural framework, it guides the reader through the subject's core elements.
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Search for:. Author : P. These rings are examples of quotient rings, which are considered in detail in chapter For Mat 2, Z the zero and unity are 0 0 1 0 and. The set of functions is a ring under these operations.
This ring has no unity. More generally, for each natural number n the set nZ of integer multiples of n is a ring. Similar rings may be defined with other square-free integers in place of 3. This is the analogue of the ring of integers in the field of complex numbers. We now present some basic facts about rings.
It is useful to establish these properties for rings in general because we then know that these hold for any particular example of a ring that we wish to work with. Proof: Suppose that there are a pair of elements 0 1 and 0 2 in R, which are both zeroes. The result for the unity is similar and is left as an exercise.
These are called the additive cancellation laws. Proof: We prove i. By commutativity of addition ii follows. Not every ring has multiplicative cancellation laws. To carry out multiplicative cancellation a commutative ring needs to satisfy an extra axiom, the zero divisor law. This type of ring, known as an integral domain, is discussed in chapter Next we state and prove from the ring axioms of The right-hand equality is proved similarly.
By uniqueness of additive inverses from Convince yourself that each of the examples a through h on pages satisfy the ring axioms of You may assume that these axioms hold for the rings Z and R.
Prove that any ring has at most one unity. Hint: Suppose that there are two. Let R be a ring. Prove the right hand equalities of Such a ring is known as a Boolean ring. State which axiom you are using at each step. What is the characteristic of this ring?
It should therefore come as no surprise that along with rings comes the concept of a subring. We begin with the definition: Notice that a subring has the same binary operations as the ring that it lies inside. Examples a Z is a subring of Q, and Q is a subring of R. Notice that for any ring R, the trivial ring 0 and R itself are subrings of R. Examples a Z is a subset of Q. Hence Z is a subring of Q, by We now introduce a special type of subring known as an ideal, which is important in the chapters that follow.
Ideals play a role in ring theory analogous to the role of normal subgroups in group theory. Notice that in any ring R the trivial ring 0 and R itself are ideals. We call an ideal that is not equal to R a proper ideal.
Every ideal is a subring, but not every subring is an ideal. Examples a For any natural number n the subset nZ is an ideal in Z. The following fact arises from Euclid's algorithm: For integers a and b we write a b if a divides b exactly without remainder. Proof: Suppose that p divides ab but does not divide a. By induction, a corollary of The equivalence classes are known as cosets.
As in the case of groups, a homomorpism is a mapping which preserves structure. But whereas with a group there is only a single binary operation to preserve, for rings we need to preserve two binary operations. This leads us to the following definition: Monomorphisms, epimorphisms and isomorphisms are defined exactly as for groups: If f is injective then it is called a monomorphism.
If f is surjective then it is called an epimorphism. If f is bijective both injective and surjective then it is called an isomorphism. Where a pair of rings are isomorphic, they may be thought of as essentially the same ring.
Hence f is a ring homomorphism. Hence f is injective and so is a ring monomorphism. In this way, we may think of the complex numbers as the vector space R 2 equipped with a suitable multiplication. This is the second way of defining the complex numbers without making use of the square root of minus one. We shall meet yet a third when we study quotient rings in chapter We denote the zero elements of rings R and S by 0 R and 0 S respectively.
Hence by the uniqueness of additive inverses from These are similar to the definitions of image and kernel of a group homomorphism from 2. Also by Hence by Convince yourself that each of the examples a through f of Q is isomorphic to the ring of quaternions, explored in chapter Find all of the ideals in the ring Z Suppose that I and J are ideals in a ring R.
Let S be the ring of The construction is applicable to any ring, so that we can say for example what it means to perform arithmetic modulo a polynomial. Here is the criterion for two cosets being equal: Next, we show that distinct cosets are disjoint: The verification of the ring axioms is tedious but easy. Hence q is an epimorphism. Examples a 3Z is an ideal in Z. The following important theorem allows us to prove isomorphism between quotient rings, arising from the construction above, and other familiar rings.
Proof: The proof is presented in four sections: I ker f is an ideal in R: We have already shown in II The mapping f is well-defined: We show that f is independent of choice of representatives of cosets.
Hence f is also surjective. This gives us our third definition of the complex numbers, this time as a quotient ring of polynomials. We conclude the chapter with two further isomorphism theorems. The result follows by the first isomorphism theorem, We show how each of these may arise as a quotient ring. Examples a Z, Q, R and C are integral domains. So we have another way of characterising an integral domain: an integral domain is a commutative ring with unity that has no divisors of zero. Hence x is not a divisor of zero.
The lack of divisors of zero is necessary and sufficient to carry out multiplicative cancellation, as the following result shows: Hence R is an integral domain. Proof: Clearly Z is an integral domain with characteristic 0.
Hence char D must be prime. We recall the definition of a field: It is left as an exercise to show that the multiplicative inverse of each non-zero element is unique. Examples a The number systems R, Q and C are the familiar examples of fields. Z is not a field. Hence F is an integral domain. Note that the converse is false: there are integral domains that are not fields. For example the ring of integers Z is an integral domain but is not a field. However, we do have the following: Consider the elements a, a 2 , a 3 , a 4 , Since D is finite these cannot all be distinct.
Hence D is a field. Proof: If n is prime then Z n is an integral domain by If n is not prime then by It turns out that a field has no proper non-trivial ideals: The converse is also true: If a commutative ring F with unity has no ideals except 0 and F itself, then F is a field. It is left as an exercise to show this. Next we define two special types of ideal known as prime ideals and maximal ideals. The quotient rings which arise from these are integral domains and fields respectively.
Hence nZ is not a prime ideal. An ideal pZ is maximal in Z iff p is prime. We may order these ideals by inclusion, as shown in the Hasse diagram below. Thus an irreducible polynomial is one that cannot be factorised as a product of polynomials of strictly smaller degree. We defer the proof of this until chapter The following lemma will be used twice: Proof: i is left as an exercise. Each maximal ideal of R is also a prime ideal.
Proof: Let M be a maximal ideal in a commutative ring R with unity. Notice that the converse is false: in a commutative ring with unity a prime ideal may not be maximal. We now state and prove two important theorems about quotient rings arising from prime and maximal ideals: Hence P is a prime ideal in R. Once again by lemma Hence no such ideal I exists, and M is a maximal ideal.
Notice that these two theorems are consistent with the facts that any maximal ideal is a prime ideal and any field is an integral domain. This is of course the field of complex numbers, C. Find all of the divisors of zero in Z8 and in Z Find all of the maximal ideals of Z Let R be a commutative ring with unity.
Prove that if R has no proper non-trivial ideals then R is a field. We are interested in the case where this quotient ring turns out to be a field. As stated in In fact, we have already seen that this field is C, the complex numbers.
Now suppose that we begin with a finite field Z p , where p is a prime, and construct the ring of polynomials Z p [x ]. Next, we choose a monic irreducible polynomial f x of degree n in Z p [x ].
Since there are p n such polynomials, the field F has p n elements. Example Consider the field Z2. We know from proposition We wish to determine the structure of this field by giving Cayley tables for addition and multiplication. We have constructed a new field with four elements. This is written as F 4. These finite fields were first studied by Galois in the early nineteenth century. Here is a remarkable fact about finite fields: This mapping is called the Frobenius automorphism.
The multiplicative group of Z p is cyclic by Proof: First, we observe that 0 is clearly a root of the polynomial. Since a polynomial of degree q cannot have more than q roots, the result follows. Draw Cayley tables for addition and multiplication in this field. By finding a generator, show that the multiplicative group of non-zero elements in F 9 is cyclic.
By finding a generator, show that the multiplicative group of non-zero elements in F 8 is cyclic. Thus p is prime if it is divisible only by 1 and by itself. Examples a The thirty smallest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, , , , and For ease of reference, we restate a key result from chapter By induction, we may prove this corollary to Proof: First we prove the existence of a factorisation into primes: if n is prime then we are done.
If a and b are prime, then we are done. If not then we repeat the process above with each of a and b, and so on. Since the numbers get smaller at each step, this process must terminate. Now by But each of these factorisations is obtained from another by changing the order of the primes, and so the factorisations are essentially the same. Examples a For the integers Z the units are —1 and 1.
We show below that these are the only units. Then b a. Hence a and b are associates. Example Consider Z[i ] with the multiplicative norm above. Hence 1, i, -1 and -i are the only units in Z[i ].
The units are —1 and 1. In the case where a polynomial has coefficients in a ring that is not a field we need to be more careful. However, it is not an irreducible in the sense of Following from Each can be obtained from another by replacing each irreducible by an associate. Unique factorisation of integers motivates us to make the following definition: Suppose further that if such an element has two such factorisations p 1 p Then D is called a unique factorisation domain, or UFD.
We shall prove this later. This terminology is rather confusing: irreducibles are what we think of in Z as prime numbers, whereas primes are elements satisfying the property of prime numbers expressed by However, irreducibles and primes coincide in this case only because in Z we have unique factorisation. Then clearly p ab, and so it follows that either p a or p b because p is a prime.
Hence p is an irreducible. Hence u is an unit. Examples a Z is a PID. We prove this below. Hence E is not principal. It follows that Z[x ] is not a PID. Proof: Let I be an ideal in Z, and let a be the smallest positive integer in I. Proof: Suppose that I is an ideal in F[x ]. Let p x be a polynomial in I of least degree.
Hence I is an ideal. Hence a is an irreducible. This also proves Because a is irreducible, by If b and c are irreducible, then we are done. If not then we repeat the process above for each of b and c. Since we are working in a PID, by Since p 1 divides a we have p 1 divides q 1 q Hence p 1 divides q i for some i. Examples a We have shown in By the theorem, Z is a UFD.
By the theorem, [ ] F x is a UFD. Note that the converse of Clearly i is satisfied. We already knew this from The converse of We conclude with an important theorem. The proof is rather long and so is omitted. This concludes our study of rings. In the next chapter, our attention returns to vector spaces.
Find all of the units in each of the following rings: i Z10 ii Z12 iii Z14 iv Z15 For the remaining elements, state which are associates of each other.
Decide which of the following are irreducible in Z[i ]: 5 7 11 13 4. Prove that in a principal ideal domain, every prime ideal is a maximal ideal. We will define an algebra to be a vector space, which has the extra structure of a product or multiplication. We begin by reviewing some basic ideas about real vector spaces.
Examples a R 2 is a real vector space. We will prefer to view Mat 2, C as a real vector space of dimension 8. Recall from chapter 4 that a subset U of a vector space V is a subspace of V if U is itself a real vector space under the same operations of vector addition and scalar multiplication of V.
A linear transformation that is bijective is called an isomorphism. You should check this. Again, you should check this. Suppose that V is a real vector space. This is why bases are important. If a vector space V has a basis with a finite number of vectors then all bases for V have the same number of vectors.
The number of vectors in each basis of V is called the dimension, written dim V. This is an example of a vector space of infinite dimension.
The field of complex numbers already has a bilinear product, and so is a real algebra. However, this real algebra is not isomorphic to that of example a. Examples a , b and d above are commutative, whereas c and e are not. B is a subalgebra of A if B is itself a real algebra under the same operations as A. Conditions i and ii ensure that B is a subspace of A. Condition iii ensures that B is closed under the product. An algebra map that is bijective is called an isomorphism. Convince yourself that each of the following is a real vector space under the operations given in the chapter: R2 C Mat 2, R R[x ] 2.
State the dimension of each of the following real vector spaces: i R4 ii Mat 3, R iii Mat 2, C Write down a basis for each of these real vector spaces. Which of the following are subspaces of Mat 2, R? Which are subalgebras? Consider the vector space R 3 with basis e1, e2, e3.
This product is assumed to be associative. This observation will be useful to us in the sections that follow. Notice also that Ae i gives the ith column of the matrix A.
This suggests the following definition: Of course, there are easier methods to calculate the determinant. But the definition via the exterior algebra gives us powerful means of proving results about determinants.
In particular, notice that the determinant is the wedge product of the columns of the matrix. Here is a selection of results.
This is called a Laplace expansion. Here are some properties of the trace: Proof: i Using the subscript notation of We notice that the coefficient of x 2 is the trace and the constant is the determinant of A.
This suggests that we ought to be able to make sense of the other coefficients in terms of operators A i. Lie algebras are not in general associative. Examples a Consider R 3 with basis i, j, k. This is known as the general linear Lie algebra, and is denoted gl n, R. It is of dimension n 2.
Everything you wanted to know about abstract algebra, but were afraid to buy. Note: The Annual Edition has been finalized. See the note about the various Editions and changes. The current edition is for the —22 academic year, with only minor modifications to the content from the previous year's edition. It also verifies and updates substantial material about Sage to Version 9.
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