抽象代数理论与应用 - Sage练习

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"Abstract Algebra - 2016 (aata-20160809-sage-7.3) - 计算机科学" 是一本关于抽象代数的理论与应用的教材，由Thomas W. Judson（来自Stephen F. Austin State University）编写，并由Robert A. Beezer（来自University of Puget Sound）提供了相关的Sage数学软件练习。该资源的日期是2016年8月9日，且可在abstract.pugetsound.edu网站上找到。文档遵循GNU Free Documentation License 1.2或更高版本，允许自由复制、分发和修改。主要内容介绍了抽象代数的理论和应用，这是一门涵盖群论、环论、域论等核心概念的数学分支。作者Thomas W. Judson和Robert A. Beezer在教材中结合了理论教学与实践应用，通过Sage软件提供练习，帮助学生深入理解和掌握抽象代数的概念。Sage是一个开源的数学计算软件，它支持符号计算、数值计算、图形绘制等功能，非常适合用于代数的实验学习。教材还包含了一组审阅者的名单，他们对教材内容提出了有益的评论和建议，这些审阅者来自不同大学，包括University of Tennessee, Knoxville的David Anderson、University of Puget Sound的Robert Beezer、California Polytechnic State University的Myron Hood等。这些专家的参与确保了教材的质量和学术性。在学习抽象代数时，学生将接触到以下知识点： 1. **群论**：学习群的基本性质，如封闭性、交换性和逆元的存在。理解群作用的概念，以及群的同态和同构。此外，还会探讨循环群、置换群、自由群和有限生成群。 2. **环论**：研究环的定义、加法群和乘法运算，以及环的理想和商环。会介绍整环、有理数环、多项式环以及它们的特性。 3. **域论**：深入研究域，特别是了解域内的除法定理和最大公因子与最小公倍数。还将探索域上的线性代数，如向量空间、线性变换和特征值。 4. **模论**：学习模的定义、模的直和、同态和同构，以及分配律和模的结构定理。 5. **格论**：理解半序集、格的定义，格的上确界和下确界，以及格的运算性质。 6. **Galois理论**：研究域扩张与Galois群，以及如何解决代数方程的根的代数问题。 7. **矩阵理论**：探讨矩阵的加法、乘法和逆矩阵，以及行列式、秩和特征值。 8. **同调代数**：了解同调的概念，以及它在拓扑、代数几何等领域中的应用。 9. **编码理论**：利用群论和环论的概念来设计和分析纠错编码，如汉明码和循环码。通过这本教材，读者不仅可以掌握抽象代数的基础知识，还能借助Sage进行实际操作，从而加深对抽象概念的理解。同时，由于其开放许可性质，该教材可广泛传播和定制，满足不同教学环境的需求。

2 CHAPTER 1. PRELIMINARIES

may vary widely: proofs can be addressed to another student, to a professor, or to the

reader of a text. If more detail than needed is presented in the proof, then the explanation

will be either long-winded or poorly written. If too much detail is omitted, then the proof

may not be convincing. Again it is important to keep the audience in mind. High school

students require much more detail than do graduate students. A good rule of thumb for an

argument in an introductory abstract algebra course is that it should be written to convince

one’s peers, whether those peers be other students or other readers of the text.

Let us examine dierent types of statements. A statement could be as simple as “10/5 =

2;” however, mathematicians are usually interested in more complex statements such as “If

p, then q,” where p and q are both statements. If certain statements are known or assumed

to be true, we wish to know what we can say about other statements. Here p is called

the hypothesis and q is known as the conclusion. Consider the following statement: If

+ bx + c = 0 and a = 0, then

x =

−b ±

√

− 4ac

The hypothesis is ax

+ bx + c = 0 and a = 0; the conclusion is

x =

−b ±

√

− 4ac

Notice that the statement says nothing about whether or not the hypothesis is true. How-

ever, if this entire statement is true and we can show that ax

+ bx + c = 0 with a = 0 is

true, then the conclusion must be true. A proof of this statement might simply be a series

of equations:

+ bx + c = 0

x = −

x +

(

)

(

)

−

(

x +

)

− 4ac

x +

√

− 4ac

x =

−b ±

√

− 4ac

If we can prove a statement true, then that statement is called a proposition. A

proposition of major importance is called a theorem. Sometimes instead of proving a

theorem or proposition all at once, we break the proof down into modules; that is, we

prove several supporting propositions, which are called lemmas, and use the results of

these propositions to prove the main result. If we can prove a proposition or a theorem,

we will often, with very little eort, be able to derive other related propositions called

corollaries.

Some Cautions and Suggestions

There are several dierent strategies for proving propositions. In addition to using dierent

methods of proof, students often make some common mistakes when they are rst learning

how to prove theorems. To aid students who are studying abstract mathematics for the

1.2. SETS AND EQUIVALENCE RELATIONS 3

rst time, we list here some of the diculties that they may encounter and some of the

strategies of proof available to them. It is a good idea to keep referring back to this list as

a reminder. (Other techniques of proof will become apparent throughout this chapter and

the remainder of the text.)

• A theorem cannot be proved by example; however, the standard way to show that a

statement is not a theorem is to provide a counterexample.

• Quantiers are important. Words and phrases such as only, for all, for every, and for

some possess dierent meanings.

• Never assume any hypothesis that is not explicitly stated in the theorem. You cannot

take things for granted.

• Suppose you wish to show that an object exists and is unique. First show that there

actually is such an object. To show that it is unique, assume that there are two such

objects, say r and s, and then show that r = s.

• Sometimes it is easier to prove the contrapositive of a statement. Proving the state-

ment “If p, then q” is exactly the same as proving the statement “If not q, then not

p.”

• Although it is usually better to nd a direct proof of a theorem, this task can some-

times be dicult. It may be easier to assume that the theorem that you are trying

to prove is false, and to hope that in the course of your argument you are forced to

make some statement that cannot possibly be true.

Remember that one of the main objectives of higher mathematics is proving theorems.

Theorems are tools that make new and productive applications of mathematics possible. We

use examples to give insight into existing theorems and to foster intuitions as to what new

theorems might be true. Applications, examples, and proofs are tightly interconnected—

much more so than they may seem at rst appearance.

1.2 Sets and Equivalence Relations

Set Theory

A set is a well-dened collection of objects; that is, it is dened in such a manner that we

can determine for any given object x whether or not x belongs to the set. The objects that

belong to a set are called its elements or members. We will denote sets by capital letters,

such as A or X; if a is an element of the set A, we write a ∈ A.

A set is usually specied either by listing all of its elements inside a pair of braces or

by stating the property that determines whether or not an object x belongs to the set. We

might write

X = {x

, x

, . . . , x

}

for a set containing elements x

, x

, . . . , x

X = {x : x satises P}

if each x in X satises a certain property P. For example, if E is the set of even positive

integers, we can describe E by writing either

E = {2, 4, 6, . . .} or E = {x : x is an even integer and x > 0}.

4 CHAPTER 1. PRELIMINARIES

We write 2 ∈ E when we want to say that 2 is in the set E, and −3 /∈ E to say that −3 is

not in the set E.

Some of the more important sets that we will consider are the following:

N = {n : n is a natural number} = {1, 2, 3, . . .};

Z = {n : n is an integer} = {. . . , −1, 0, 1, 2, . . .};

Q = {r : r is a rational number} = {p/q : p, q ∈ Z where q = 0};

R = {x : x is a real number};

C = {z : z is a complex number}.

We can nd various relations between sets as well as perform operations on sets. A set

A is a subset of B, written A ⊂ B or B ⊃ A, if every element of A is also an element of B.

For example,

{4, 5, 8} ⊂ {2, 3 , 4, 5, 6, 7, 8, 9}

and

N ⊂ Z ⊂ Q ⊂ R ⊂ C.

Trivially, every set is a subset of itself. A set B is a proper subset of a set A if B ⊂ A but

B = A. If A is not a subset of B, we write A ⊂ B; for example, {4, 7, 9} ⊂ {2, 4, 5, 8, 9}.

Two sets are equal, written A = B, if we can show that A ⊂ B and B ⊂ A.

It is convenient to have a set with no elements in it. This set is called the empty set

and is denoted by ∅. Note that the empty set is a subset of every set.

To construct new sets out of old sets, we can perform certain operations: the union

A ∪ B of two sets A and B is dened as

A ∪ B = {x : x ∈ A or x ∈ B};

the intersection of A and B is dened by

A ∩ B = {x : x ∈ A and x ∈ B}.

If A = {1, 3, 5} and B = {1, 2, 3, 9}, then

A ∪ B = {1, 2, 3, 5, 9} and A ∩ B = {1, 3}.

We can consider the union and the intersection of more than two sets. In this case we write

∪

i=1

= A

∪ . . . ∪ A

and

∩

i=1

= A

∩ . . . ∩ A

for the union and intersection, respectively, of the sets A

, . . . , A

When two sets have no elements in common, they are said to be disjoint; for example,

if E is the set of even integers and O is the set of odd integers, then E and O are disjoint.

Two sets A and B are disjoint exactly when A ∩ B = ∅.

Sometimes we will work within one xed set U, called the universal set. For any set

A ⊂ U , we dene the complement of A, denoted by A

′

, to be the set

′

= {x : x ∈ U and x /∈ A}.

6 CHAPTER 1. PRELIMINARIES

1. (A ∪ B)

′

= A

′

∩ B

′

;

2. (A ∩ B)

′

= A

′

∪ B

′

Proof. (1) If A∪B = ∅, then the theorem follows immediately since both A and B are the

empty set. Otherwise, we must show that (A ∪ B)

′

⊂ A

′

∩ B

′

and (A ∪ B)

′

⊃ A

′

∩ B

′

. Let

x ∈ (A ∪B)

′

. Then x /∈ A ∪B. So x is neither in A nor in B, by the denition of the union

of sets. By the denition of the complement, x ∈ A

′

and x ∈ B

′

. Therefore, x ∈ A

′

∩ B

′

and we have (A ∪ B)

′

⊂ A

′

∩ B

′

To show the reverse inclusion, suppose that x ∈ A

′

∩ B

′

. Then x ∈ A

′

and x ∈ B

′

, and

so x /∈ A and x /∈ B. Thus x /∈ A ∪ B and so x ∈ (A ∪B)

′

. Hence, (A ∪ B)

′

⊃ A

′

∩ B

′

and

so (A ∪ B)

′

= A

′

∩ B

′

The proof of (2) is left as an exercise.

Example 1.4. Other relations between sets often hold true. For example,

(A \ B) ∩ (B \ A) = ∅.

To see that this is true, observe that

(A \ B) ∩ (B \ A) = (A ∩ B

′

) ∩ (B ∩ A

′

)

= A ∩ A

′

∩ B ∩ B

′

= ∅.

Cartesian Products and Mappings

Given sets A and B, we can dene a new set A × B, called the Cartesian product of A

and B, as a set of ordered pairs. That is,

A × B = {(a, b) : a ∈ A and b ∈ B}.

Example 1.5. If A = {x, y}, B = {1, 2, 3}, and C = ∅, then A × B is the set

{(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)}

and

A × C = ∅.

We dene the Cartesian product of n sets to be

× ··· × A

= { (a

, . . . , a

) : a

∈ A

for i = 1, . . . , n}.

If A = A

= A

= ··· = A

, we often write A

for A × ··· × A (where A would be written

n times). For example, the set R

consists of all of 3-tuples of real numbers.

Subsets of A×B are called relations. We will dene a mapping or function f ⊂ A×B

from a set A to a set B to be the special type of relation where (a, b) ∈ f if for every element

a ∈ A there exists a unique element b ∈ B. Another way of saying this is that for every

element in A, f assigns a unique element in B. We usually write f : A → B or A

→ B.

Instead of writing down ordered pairs (a, b) ∈ A × B, we write f(a) = b or f : a 7→ b. The

set A is called the domain of f and

f(A) = {f (a) : a ∈ A} ⊂ B

is called the range or image of f . We can think of the elements in the function’s domain

as input values and the elements in the function’s range as output values.

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