Pages: 136

No chapter abstract available.

Pages: 9

A semigroup is a set M with a binary operation https://www.w3.org/1998/Math/MathML" display="inline"> ⋅ https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math1_1.tif"/> satisfying (M1) (associativity) https://www.w3.org/1998/Math/MathML" display="inline"> ( a b ) c = a ( b c ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math1_2.tif"/>

Pages: 10

Our goal in this lesson is to introduce subgroups and show that the order of a subgroup divides the order of the group. This raises a basic question in arithmetic. How do we verify that m divides n (written https://www.w3.org/1998/Math/MathML" display="inline"> m   |   n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math2_1.tif"/> )? To avoid confusion, in this chapter m and n will denote positive numbers, and p a positive prime number. Two methods are already available from elementary school: Euclidean division says that https://www.w3.org/1998/Math/MathML" display="inline"> n = q m + r https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math2_2.tif"/> for a suitable quotient q and remainder r with https://www.w3.org/1998/Math/MathML" display="inline"> 0 ≤ r < m https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math2_3.tif"/> , so one need merely check https://www.w3.org/1998/Math/MathML" display="inline"> r = 0 https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math2_4.tif"/> . Possibly even more fundamentally, https://www.w3.org/1998/Math/MathML" display="inline"> m   |   n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math2_5.tif"/> iff one can partition a set of n elements into subsets each having m elements.

Pages: 7

In Chapter 2 we saw that Lagrange's theorem has the consequence that if https://www.w3.org/1998/Math/MathML" display="inline"> g ∈ G https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_1.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> o ( g ) = m https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_2.tif"/> , then m divides https://www.w3.org/1998/Math/MathML" display="inline"> | G | https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_3.tif"/> . This raises the converse question: “If m divides https://www.w3.org/1998/Math/MathML" display="inline"> | G | https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_4.tif"/> then does https://www.w3.org/1998/Math/MathML" display="inline"> o ( g ) = m https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_5.tif"/> for suitable g in G?” Our goal will be to prove this for m prime; for m not prime the claim is false, as evidenced by https://www.w3.org/1998/Math/MathML" display="inline"> Euler ( 8 ) = { 1 , 3 , 5 , 7 } https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math3_6.tif"/> , which has order 4 although each of its elements has order 2. In proving Lagrange's theorem we examined the process of division. In studying the converse we shall learn how to count.

Pages: 6

Having proved some rather deep theorems about subgroups, let us pause and take stock of what we know about specific groups. We have also seen groups arising in other contexts, such as permutations of sets of objects, and invertible linear transformations of vector spaces. Groups also arise naturally in geometry. Define a symmetry of a geometric object to be a transformation of the object which preserves distances. For example, any symmetry of a circle is either a rotation or a flip. The set of symmetries of any given solid is a group.

Pages: 12

We have seen that the kernel of any group homomorphism https://www.w3.org/1998/Math/MathML" display="inline"> φ   :   G → H https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math5_1.tif"/> is a subgroup of G. Our object here is to study a special property of this kind of subgroup, which turns out to be the key to the structure theory.

Pages: 11

The classification problem for a family of groups is the question of how to list all groups in this family “up to isomorphism,” i.e., one representative for each class of isomorphic groups. Having such a list, presumably we could verify assertions about groups simply by going down the list. However, there are several difficulties with this approach when applied to large families, such as all finite groups. First of all, the list might be infinite, since https://www.w3.org/1998/Math/MathML" display="inline"> ( Z m , + ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math6_1.tif"/> is a finite group for each m. Moreover, even if we had a complete list of finite groups, it might be impossible in practice to verify any given assertion. For example, https://www.w3.org/1998/Math/MathML" display="inline"> 1 , 2 , 3 , … https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math6_2.tif"/> is a list of the natural numbers, but we do not know if there are any odd perfect numbers. Finally, the list of finite groups might be less enlightening than the body of theorems used to obtain the list. Nevertheless, the classification problem is the focal research problem in the theory of finite groups, and the ensuing results have been of great use.

Pages: 12

Abelian groups have a much more manageable structure than groups in general. We shall see soon that finite abelian groups are not much more complicated than cyclic groups, but first let us illustrate their nice structure by means of some easy facts.

Pages: 12

In contrast to the abelian case, generation in arbitrary groups is very complicated; we always need some other ingredient to describe a group, namely the relations among the generators, i.e., those expressions among the generators that are equal in the group.

Pages: 14

We have just constructed several new candidates for groups, e.g., certain semidirect products and the quaternion group of order 8, and have shown by various ad hoc arguments in the exercises that they are indeed groups. Several of these proofs involved displaying the proposed group as a subgroup of Sn (cf. Exercise 8.9) or even a larger group (Remark 35 and Exercise 8.6). We would like to see why these proofs work, and in doing so shall develop a general procedure that is guaranteed to work for any group.

Pages: 8

Recall that an element b is a conjugate of a in a group G if https://www.w3.org/1998/Math/MathML" display="inline"> b = g a g − 1 https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math10_1.tif"/> for suitable g in G. Having seen already in Chapter 5 how conjugacy arises in determining when a subgroup is normal, we want to study conjugacy now in its own right.

Pages: 7

In the discussion of finite abelian groups we saw that every abelian group is isomorphic to the direct product of its Sylow subgroups. Of course, this fails terribly for most nonabelian groups, but amazingly enough, one can salvage a good part of this theorem. In the process we shall come up with a powerful new invariant—the number of p-Sylow subgroups. Our basic tool is a generalization of conjugacy classes. First we generalize centralizers of elements.

Pages: 11

There is a certain class of groups, which is considerably more general than abelian groups, but possess some of their nice properties. We shall discuss these groups now, in preparation for an application for the solution of polynomial equations in Chapter 27. The underlying question is, “How far is a given group from being abelian?” The answer lies in the rather trivial fact that https://www.w3.org/1998/Math/MathML" display="inline"> a b = b a https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math12_1.tif"/> iff https://www.w3.org/1998/Math/MathML" display="inline"> a b a − 1 b − 1 = e https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math12_2.tif"/> , which we needed in Proposition 6.14. This leads us to view the subset of elements of the form https://www.w3.org/1998/Math/MathML" display="inline"> a b a − 1 b − 1 https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math12_3.tif"/> as the obstruction to a group being abelian.

Pages: 11

Groups of matrices, which are most often infinite, are of utmost importance in mathematics. Recall that https://www.w3.org/1998/Math/MathML" display="inline"> GL n ( F ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math13_1.tif"/> denotes the group of invertible https://www.w3.org/1998/Math/MathML" display="inline"> n × n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math13_2.tif"/> matrices over a field F. In particular https://www.w3.org/1998/Math/MathML" display="inline"> GL 1 ( F ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math13_3.tif"/> is the multiplicative group of the field, often denoted F ×.

Pages: 4

True or false (Prove or give a counterexample): The set of left invertible elements in a monoid forms a group. If https://www.w3.org/1998/Math/MathML" display="inline"> a b = e https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math_1.tif"/> , then https://www.w3.org/1998/Math/MathML" display="inline"> b a = e https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math_2.tif"/> .

Pages: 72

No chapter abstract available.

Pages: 7

In this chapter we lay out the foundations of rings, focusing on domains and (skew) fields; in the next chapter, we complement the discussion with the theory of homomorphisms and ideals.

Pages: 8

Although our emphasis in group theory was on finite groups, our attention here is mainly on infinite rings, because many of the rings of greatest interest to us are infinite, such as https://www.w3.org/1998/Math/MathML" display="inline"> Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math15_1.tif"/> , https://www.w3.org/1998/Math/MathML" display="inline"> Q https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math15_2.tif"/> , https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math15_3.tif"/> , and https://www.w3.org/1998/Math/MathML" display="inline"> C https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math15_4.tif"/> . (Finite fields also are of interest, and will be classified in Chapter 25.) However, the classification problem for infinite rings is so formidable as to be nigh impossible. In providing manageable partial results, the structure theory for rings assumes an important role and pervades ring theory. After a while the structure theory takes on a form of its own, guiding us to the most profitable avenues of inquiry. We avail ourselves of the structure theory of groups, making use of the fact that every ring is an abelian group (under +). Thus, at the outset, we pass through the following stages: ring homomorphism, kernel of homomorphism ↔ ideals (analogous to normal subgroups), residue ring = factor ring https://www.w3.org/1998/Math/MathML" display="inline"> R / A https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math15_5.tif"/> where A is an ideal of R, structure of https://www.w3.org/1998/Math/MathML" display="inline"> R / A https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math15_6.tif"/> in terms of R, and Noether isomorphism theorems.

Pages: 5

We have seen that any subring of a field is an integral domain, and every finite integral domain is a field. On the other hand, although https://www.w3.org/1998/Math/MathML" display="inline"> Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math16_1.tif"/> is an integral domain that is not a field, https://www.w3.org/1998/Math/MathML" display="inline"> Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math16_2.tif"/> is a subring of the field https://www.w3.org/1998/Math/MathML" display="inline"> Q https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math16_3.tif"/> . This leads us to ask whether every integral domain need be a subring of a suitable field. Actually, the standard construction of https://www.w3.org/1998/Math/MathML" display="inline"> Q https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math16_4.tif"/> from https://www.w3.org/1998/Math/MathML" display="inline"> Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math16_5.tif"/> can be modified to yield a positive response to this query.

Pages: 12

As indicated in the introduction to Part II, some of the most beautiful results of algebra involve polynomials. Upon first acquaintance, a polynomial seems rather ethereal – a formal expression involving an indeterminate x and coefficients from a given ring R – and one might expect that we must substitute some value for x in order to obtain meaningful results. However, it turns out that the collection of all these polynomials can be given the structure of a ring having many nice properties which are inherited from R, and in this sublime transition from chaos to algebraic structure, x becomes a very meaningful element of the new ring.

Pages: 11

Although the applications given in the previous section are quite beautiful, the basic theorems on unique factorization might seem somewhat ad hoc, based on the fortuitous appearance of Euclidean division and the degree function, and are rather cumbersome to state. Actually one can rework the arithmetic theory more intrinsically in terms of ideals, so that many of the main results become easier to state, and in greater generality.

Pages: 7

This chapter (and much of our subsequent material) grows out of the following innocuous application of Euclidean division for polynomials: Remark 1 (Notation as in <xref ref-type="boxed-text" rid="prop17_7">Proposition 17.7</xref>.) When R is commutative and https://www.w3.org/1998/Math/MathML" display="inline"> g = x − c https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_1.tif"/> with https://www.w3.org/1998/Math/MathML" display="inline"> c ∈ R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_2.tif"/> , we can calculate r directly, by means of “substituting” c for x; since r is constant and https://www.w3.org/1998/Math/MathML" display="inline"> g ( c ) = 0 https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_3.tif"/> , 19.1 https://www.w3.org/1998/Math/MathML" display="block"> r = f ( c ) − q ( c ) g ( c ) = f ( c ) . https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_4.tif"/> Although the reader might be willing to accept this remark without further ado, let us be careful and justify the substitution we have just made, by proving Lemma 2 (“Substitution lemma”). Suppose R is commutative. For any given https://www.w3.org/1998/Math/MathML" display="inline"> c ∈ R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_5.tif"/> , there is a “substitution” homomorphism https://www.w3.org/1998/Math/MathML" display="inline"> R [ x ] → R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_6.tif"/> defined by https://www.w3.org/1998/Math/MathML" display="inline"> ∑ a i x i ↦ ∑ a i c i https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_7.tif"/> . (In other words we “substitute” c for x.) Actually the following more general result will also be useful (One recovers Lemma 2 by taking C = R and https://www.w3.org/1998/Math/MathML" display="inline"> ψ = 1 R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_8.tif"/> ). Lemma 3 Suppose C is a commutative ring, and https://www.w3.org/1998/Math/MathML" display="inline"> c ∈ C https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_9.tif"/> . Any given ring homomorphism https://www.w3.org/1998/Math/MathML" display="inline"> ψ   :   R → C https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_10.tif"/> extends uniquely to a ring homomorphism https://www.w3.org/1998/Math/MathML" display="inline"> φ   :   R [ x ] → C https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_11.tif"/> , such that https://www.w3.org/1998/Math/MathML" display="inline"> φ ( x ) = c https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_12.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> φ ( a ) = ψ ( a ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_13.tif"/> for all a in R. Proof. If such φ exists, we must have 19.2 https://www.w3.org/1998/Math/MathML" display="block"> φ ( ∑ a i x i ) = ∑ φ ( a i ) φ ( x i ) = ∑ ψ ( a i ) c i , https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_14.tif"/> proving uniqueness. It remains to show that φ as defined by (19.2) is a homomorphism. https://www.w3.org/1998/Math/MathML" display="block"> φ ( ∑ a i x i + ∑ b i x i ) = φ ( ∑ ( a i + b i ) x i ) = ∑ ψ ( a i + b i ) c i = ∑ ψ ( a i ) c i + ∑ ψ ( b i ) c i = φ ( ∑ a i x i ) + φ ( ∑ b i x i ) ; https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_15.tif"/> https://www.w3.org/1998/Math/MathML" display="block"> φ ( ( ∑ i a i x i )   ( ∑ j b j x j ) ) = φ ( ∑ u ∑ i = 0 u a i b u − i x u ) = ∑ u ∑ i = 0 u ψ ( a i ) ψ ( b u − i ) c u = ( ∑ i ψ ( a i ) c i )   ( ∑ j ψ ( b j ) c j ) = φ ( ∑ i a i x i ) φ   ( ∑ j b j x j ) . https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_16.tif"/> (Here the hypothesis C commutative was needed to move c past https://www.w3.org/1998/Math/MathML" display="inline"> ψ ( b j ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_17.tif"/> .) Finally, https://www.w3.org/1998/Math/MathML" display="inline"> φ ( 1 ) = ψ ( 1 ) = 1 https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math19_18.tif"/> . □

Pages: 9

We are ready to tackle number theory in Euclidean rings other than https://www.w3.org/1998/Math/MathML" display="inline"> Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math20_1.tif"/> . Our motivation is that polynomials such as https://www.w3.org/1998/Math/MathML" display="block"> x n − 1 = ( x − 1 )   ( x n − 1 + x n − 2 + ⋯ + 1 ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math20_2.tif"/>

Pages: 9

Having seen that irreducible polynomials play a special role in the polynomial ring F[x], we should like to be able to determine which polynomials are irreducible. Later on we shall see that every irreducible polynomial gives rise to a field, thereby enhancing our interest in irreducible polynomials.

Pages: 2

Prove or give a counterexample: The union of two ideals is an ideal.

Pages: 76

No chapter abstract available.

Pages: 9

We start with a given base field F containing all the coefficients of the equation to be studied, and build up from there.

Pages: 12

In the eyes of the ancient Greeks, there was a deep connection between mathematics and life, and numbers entered into every facet of philosophy, science, and art, such as the “golden ratio” https://www.w3.org/1998/Math/MathML" display="inline"> 1 + 5 2 https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math23_1.tif"/> , in architecture. The only real numbers of interest to them could be constructed, using a compass and straight edge, by means of certain well-defined rules, which in modern terms could be described as follows:

Pages: 13

In the previous sections we studied algebraic elements in terms of the polynomials that they satisfy. Now we reverse the question and ask, how can we determine the roots of a given polynomial? This question has several possible interpretations, and there exists a highly developed theory of approximating roots of a polynomial in a given field. However, we are interested in an exact algebraic solution, and thus our first interpretation is, “Given https://www.w3.org/1998/Math/MathML" display="inline"> f ∈ F [ x ] https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math24_1.tif"/> , can we find a field K that contains all the roots of f?” First we produce one root. To understand the question, let us examine a familiar situation.

Pages: 5

Since the beginning of computer technology, the field https://www.w3.org/1998/Math/MathML" display="inline"> F 2 https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math25_1.tif"/> has had important applications, with “1” denoting a closed circuit and “0” denoting an open circuit. Since computers use more than one circuit, a natural and important question is whether there are fields whose orders are higher powers of 2, or of other orders. So far we know https://www.w3.org/1998/Math/MathML" display="inline"> F p https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math25_2.tif"/> is a field for p prime, so, in particular, there are fields of order 2,3,5,7, and so forth. What about 4 and 6? More generally, we want to tackle the following two questions, given n in https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math25_3.tif"/> : Is there a field of n elements? Are there two nonisomorphic fields of n elements?

Pages: 10

One of the most beautiful achievements of mathematics is Galois' theory, linking field extensions to groups and thereby enabling one to apply group theory to study field extensions. Although this theory can fill a course by itself, we shall focus on the main results, in the following three chapters.

Pages: 8

We are about to embark on a series of fascinating applications in algebra, representing some of the high points in the history of mathematics. The underlying philosophy is to use the Galois correspondence to translate questions about intermediate fields to parallel questions in group theory, and then to use known results about groups (from Part I).

Pages: 13

In this chapter we present the crowning application of Galois theory: the determination, in terms of group theory, of the solvability of equations by means of formulas involving nth roots. To make life easier, we shall assume throughout that F is a perfect field, so that we do not need to worry about separability.

Pages: 2

True or false: https://www.w3.org/1998/Math/MathML" display="inline"> C https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math104_1.tif"/> is the algebraic closure of https://www.w3.org/1998/Math/MathML" display="inline"> Q https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math104_2.tif"/> .

Pages: 56

No chapter abstract available.

Pages: 3

Earlier we considered the ring https://www.w3.org/1998/Math/MathML" display="inline"> Z [ ζ ] https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math29_1.tif"/> in conjunction with number theory and Galois theory. Now we study https://www.w3.org/1998/Math/MathML" display="inline"> Z [ ζ ] https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math29_2.tif"/> in its own right, by generalizing algebraicity over a field. Suppose throughout that https://www.w3.org/1998/Math/MathML" display="inline"> R ⊇ C https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math29_3.tif"/> are integral domains, in which case we say that R is an extension of C.

Pages: 11

Group representations were introduced in Remark 9.34 mainly to prove the existence of certain finite groups. In this chapter we bring in group representation as a powerful tool which enables us to prove Burnside's Theorem on solvability of groups.

Pages: 5

Although our effort in field theory has been concentrated in studying algebraic numbers, “most” real numbers are transcendental over https://www.w3.org/1998/Math/MathML" display="inline"> Q https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math31_1.tif"/> ! (This is in view of Exercise 22.16, because https://www.w3.org/1998/Math/MathML" display="inline"> R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math31_2.tif"/> is not countable.) Strangely enough, the task of finding even one specific transcendental number is considerably more difficult. Many of the proofs of transcendence of particular numbers rely on the fundamental theorem of calculus: If f is a continuous function on the interval [a, b] with continuous derivative f ′ on https://www.w3.org/1998/Math/MathML" display="inline"> ( a , b ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math31_3.tif"/> , then https://www.w3.org/1998/Math/MathML" display="block"> f ( b ) − f ( a ) = ( b − a ) f ′ ( c ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math31_4.tif"/>

Pages: 8

These notes have dealt almost exclusively with commutative rings, in part because of the greater ease in dealing with one-sided ideals than with two-sided ideals. However, noncommutative rings do arise naturally in many mathematical contexts, and should also be considered. In harmony with one of our themes, we shall apply certain noncommutative rings (namely skew fields) to elementary number theory, although there are several deep connections to geometry and other subjects that are beyond the scope of these notes. Our brief excursion into skew fields is motivated by the following theorem:

Pages: 27

Often one could use a generalization of vector spaces, for any base ring, not necessarily a field, satisfying the same axioms as in Definition 22.7: Definition 1 A (left) module over a ring R is an abelian group https://www.w3.org/1998/Math/MathML" display="inline"> ( M , + ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_1.tif"/> endowed with scalar multiplication https://www.w3.org/1998/Math/MathML" display="inline"> R × M → M https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_2.tif"/> satisfying the following axioms, for all ri in R and a,b in M: https://www.w3.org/1998/Math/MathML" display="block"> 1 a = a ; ( r 1 + r 2 ) a = r 1 a + r 2 a ; r ( a + b ) = r a + r b ; ( r 1 r 2 ) v = r 1 ( r 2 v ) . https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_3.tif"/> Example 2 Any ring R is a module over itself, via the ring multiplication. More generally, any left ideal of R is an R-module. The modules over a field are simply vector spaces. Any abelian group M is a module over https://www.w3.org/1998/Math/MathML" display="inline"> Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_4.tif"/> is a module, where we define https://www.w3.org/1998/Math/MathML" display="inline"> n v = v + ⋯ + v https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_5.tif"/> taken n times for https://www.w3.org/1998/Math/MathML" display="inline"> n ≥ 0 , https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_6.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> ( − n ) v = − ( n v ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_7.tif"/> . Every module over https://www.w3.org/1998/Math/MathML" display="inline"> Z https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_8.tif"/> has this form. If V is a vector space over a field F with a linear action of a group G on V, then V is an F[G]-module. If V is a vector space over a field F and https://www.w3.org/1998/Math/MathML" display="inline"> T : F → F https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_9.tif"/> is a linear transformation, then V is an F[x]-module, under the operation https://www.w3.org/1998/Math/MathML" display="inline"> ∑ ( a i λ i ) ⋅ v = ∑ a i ( T i ( v ) ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_10.tif"/> for https://www.w3.org/1998/Math/MathML" display="inline"> a i ∈ F https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_11.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> v ∈ V , https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429278655/99fc54fc-de3d-4cef-b607-11e53a6ee9fb/content/math33_12.tif"/> where T 0 is the identity map.