按：這篇筆記是系列筆記的第三篇，第一部分有4節(jié)字旭，每節(jié)對應(yīng)1-2篇筆記。
筆記的方式崖叫，是引用一段個人覺得比較有亮點的英文原文遗淳，再給一段簡化的中文說明，不采用中文版的翻譯心傀，不自行做直接翻譯屈暗，只說明要點。因為不可能大段大段地去引用脂男，必然會有語境的丟失养叛，會做一些補充說明，以“按：”開始宰翅。對中文版翻譯進行更正或調(diào)整的說明弃甥，以“注：”開始。偶爾也會插入自己的議論汁讼，以“評：”開始淆攻。

前兩篇筆記為：

• 《普林斯頓數(shù)學(xué)指引》讀書筆記——I.1 數(shù)學(xué)是關(guān)于什么的
• 《普林斯頓數(shù)學(xué)指引》讀書筆記——I.2 數(shù)學(xué)的語言和語法

I.3 Some Fundamental Mathematical Definitions （一些基本的數(shù)學(xué)定義）

1 The Main Number Systems （主要的數(shù)系）

The modern view of numbers is that they are best regarded not individually but as parts of larger wholes, called number systems; the distinguishing features of number systems are the arithmetical operations—such as addition, multiplication, subtraction, division, and extraction of roots—that can be performed on them. This view of numbers is very fruitful and provides a springboard into abstract algebra.

對于數(shù)的現(xiàn)代視角是，最好不要把數(shù)當(dāng)作獨立個體嘿架，而應(yīng)視為一個更大的整體的一部分瓶珊，這個整體稱之為數(shù)系。數(shù)系最突出的特點是耸彪，可以在其上完成算術(shù)運算伞芹，包括加、減搜囱、乘丑瞧、除柑土、開方蜀肘。這種關(guān)于數(shù)的視角是富有成果的绊汹，它是通向抽象代數(shù)的跳板。

注：中文版將“individually”譯為“孤立地”不太準(zhǔn)確扮宠，另外“view”翻譯成“視角”要比“觀點”更自然西乖。

Of course, the phrase “1, 2, 3, 4, and so on” does not constitute a formal definition, but it does suggest the following basic picture of the natural numbers, one that we tend to take for granted. (i) Given any natural number n there is another, n+1, that comes next—known as the successor of n. (ii) A list that starts with 1 and follows each number by its successor will include every natural number exactly once and nothing else. This picture is encapsulated by the Peano Axioms [III.69].

當(dāng)然，“1, 2, 3, 4 ……”這樣的描述坛增，并不算是正式的定義获雕，但它的確提出了下面這個我們視為理所當(dāng)然的對自然數(shù)的描述：

(i) 給定一個自然數(shù)n，后面必然跟著一個自然數(shù)n+1收捣，稱為n的后繼者届案；
(ii) 一個從1開始，且隨后每一個數(shù)是前一個數(shù)的后繼者的數(shù)列罢艾，會正好包含每個自然數(shù)各一次楣颠，且不再包含其他東西。

這個描述被濃縮為佩亞諾公理咐蚯。

The set of all integers—positive, negative, and zero— is usually denoted Z (for the German word “Zahlen,” meaning “numbers”). Within this system, subtraction is always possible: that is, if m and n are integers, then so is m - n.

所有整數(shù)——正負(fù)整數(shù)與零——的集合童漩，常記作Z（德文表示數(shù)的單詞“Zahlen,”的第一個字母），在這個屬性里春锋，減法總是可能的：即如果m和n都是整數(shù)矫膨，那么m-n也是。

A more theoretical justification for the rational numbers is that they form a number system in which division is always possible——except by zero. This fact, together with some basic properties of the arithmetical operations, means that Q is a field.

按：上文提到期奔，如果只需要計數(shù)侧馅，那整數(shù)就夠了，需要有理數(shù)的其中一個理由是測量的需要呐萌，包括長度馁痴、重量、溫度和速度等搁胆。

一個為有理數(shù)的更為理論化的合理性論證弥搞，是它們組成了一個除法總是可能的數(shù)系（除了除以零），這個事實渠旁，以及一些算術(shù)運算的基本性質(zhì)攀例，意味著Q是一個域。

注：這里把“justification”譯作“合理性論證”顾腊，其實也可以更簡單地譯為“依據(jù)”粤铭。因為這里作者其實是想要為引入沒那么自然的數(shù)系尋找依據(jù)，解釋其必要性和合理性杂靶。

Because real numbers are intimately connected with the idea of limits (of successive approximations), a true appreciation of the real number system depends on an understanding of mathematical analysis.

由于實數(shù)與（逐次逼近的）極限過程緊密地聯(lián)系著梆惯，對實數(shù)系真正的領(lǐng)會就依賴于對數(shù)學(xué)分析的理解酱鸭。

注：中文版將“successive approximations”譯為“逐步逼近”，但其對應(yīng)的數(shù)學(xué)術(shù)語的常見譯法是“逐次逼近”垛吗，其英文解釋“A method for estimating the value of an unknown quantity by repeated comparison to a sequence of known quantities”里的關(guān)鍵詞也是“repeated”而非“gradually”凹髓，因此此處不應(yīng)用“逐步”。

按：這里下一小節(jié)討論了復(fù)數(shù)怯屉，但沒有什么特別的洞見蔚舀，所以沒有摘錄。要真正理解復(fù)數(shù)锨络，需要理解Geometric Algebra赌躺，回頭會單獨整理筆記。

2 Four Important Algebraic Structures （四個重要的代數(shù)結(jié)構(gòu)）

按：這節(jié)簡要介紹了群羡儿、域礼患、向量空間和環(huán)。

If S is any mathematical structure, then a symmetry of S is a function from S to itself that preserves its structure. If S is a geometrical shape, then the mathematical structure that should be preserved is the distance between any two of its points.

如果S是任意的數(shù)學(xué)結(jié)構(gòu)掠归，S的對稱就是一個由S到其自身的缅叠、能保持這個結(jié)構(gòu)的函數(shù)。例如拂到，當(dāng)S是一個幾何圖形時痪署，則應(yīng)該得到保持的數(shù)學(xué)結(jié)構(gòu)（之一），就是其上任意兩點的距離兄旬。

It is fruitful to draw an analogy with the geometrical situation and regard any structure-preserving function as a sort of symmetry. Because of its extreme generality, symmetry is an all-pervasive concept within mathematics; and wherever symmetries appear, structures known as groups follow close behind.

與幾何的情況進行類比狼犯，并把任意可以保持結(jié)構(gòu)的函數(shù)都當(dāng)作某種對稱，這樣做是富有成果的领铐。由于其極度的通用性悯森，對稱是一個在數(shù)學(xué)里無處不在的概念；而且只要哪里有了對稱出現(xiàn)绪撵，像群這樣的概念就會如影隨形瓢姻。

Although several number systems form groups, to regard them merely as groups is to ignore a great deal of their algebraic structure. In particular, whereas a group has just one binary operation, the standard number systems have two, namely addition and multiplication (from which further ones, such as subtraction and division, can be derived).

雖然好幾個數(shù)域都是群，但只把他們看成群音诈，就忽略了其代數(shù)結(jié)構(gòu)很大的一部分幻碱。尤其是群里面只有一個二元運算，標(biāo)準(zhǔn)的數(shù)域卻有兩個细溅，即加法和乘法（由它們還可以得到其他附加的運算褥傍，比如減法和除法）。

A very general rule when defining mathematical structures is that if a definition splits into parts, then the definition as a whole will not be interesting unless those parts interact. Here our two parts are addition and multiplication, and the properties mentioned so far do not relate them in any way. But one final property, known as the distributive law, does this, and thereby gives fields their special character.

在定義數(shù)學(xué)結(jié)構(gòu)的時候喇聊，有一個很一般的原理：如果一個數(shù)學(xué)定義恍风，可以分成幾個部分，則除非這些部分可以相互作用，否則這個定義就沒有什么意思（僅僅相當(dāng)于分成的幾個部分對應(yīng)的原來就定義過的數(shù)學(xué)結(jié)構(gòu)而已）朋贬。域的加法和乘法凯楔，就是這樣的兩個部分，而迄今為止提到的所有性質(zhì)锦募，并未把它們以某種方式聯(lián)系起來摆屯。然而，最后的一個性質(zhì)御滩，即分配律鸥拧，做到了這一點党远，從而給了域獨有的特性削解。

In addition to Q, R, and C, one other field stands out as fundamental, namely Fp, which is the set of integers modulo a prime p, with addition and multiplication also defined modulo p (see Modular Arithmetic [III.60]).

除了Q、R沟娱、C之外氛驮，還有一個引人注目的基礎(chǔ)域，即Fp济似。它是整數(shù)對素數(shù)p取模組成的集合矫废，其中的加法、減法砰蠢，也被定義為對p取模蓖扑，詳見模算術(shù)。

There is an important process of extension that allows one to build new fields out of old ones. The idea is to start with a field F, find a polynomial P that has no roots in F, and “adjoin” a new element to F with the stipulation that it is a root of P. This produces an extended field F, which consists of everything that one can produce from this root and from elements of F using addition and multiplication.

有一個重要的過程與域有關(guān)台舱，這個過程稱之為域的擴張律杠，它使我們能夠從原來的域構(gòu)造出新的域來。其基本的思想就是從一個域F開始竞惋，找一個在F中沒有根的多項式P柜去，然后把一個新的元素附加到F上，約定這個元素就是P的根拆宛。這樣的過程嗓奢，會產(chǎn)生一個擴張的域，它會包含浑厚，所有可以用這個根與F中的元素通過加法和乘法產(chǎn)生出來所有“數(shù)”股耽。

評：這段話從抽象的角度，描述了帶來整個復(fù)數(shù)域的i（定義為x^2+1這個多項式的根）的誕生過程钳幅。

A second very significant justification for introducing fields is that they can be used to form vector spaces.

引入域的另外一個重要依據(jù)是物蝙，它們可以用來構(gòu)成向量空間。

A vector space is a mathematical structure in which the notion of linear combination makes sense.

向量空間就是一個線性組合的概念在其中有意義的數(shù)學(xué)結(jié)構(gòu)贡这。

There is one final remark to make about scalars. They were defined earlier as real numbers that one uses to make linear combinations of vectors. But it turns out that the calculations one does with scalars, in particular solving simultaneous equations, can all be done in a more general context. What matters is that they should belong to a field, so Q, R, and C can all be used as systems of scalars, as indeed can more general fields. If the scalars for a vector space V come from a field F, then one says that V is a vector space over F. This generalization is important and useful: see, for example, Algebraic Numbers [IV.3 §17].

關(guān)于標(biāo)量還有最后一個說明茬末。之前，標(biāo)量被定義為構(gòu)造向量的線性組合時所用的實數(shù)。其實丽惭，我們用標(biāo)量所做的計算击奶，尤其是在解聯(lián)立方程時，在更廣泛的語境下也可以做责掏。真正重要的是柜砾，（用于計算的“數(shù)”）必須屬于一個域，所以Q换衬、R痰驱、C都可以用作標(biāo)量的系統(tǒng)，更一般的域也是可以的瞳浦。如果一個向量空間V的標(biāo)量來自域F担映，我們就說V是域F上的向量空間，這個推廣重要而且有用叫潦，可見代數(shù)數(shù)蝇完。

Roughly speaking, a ring is an algebraic structure that has most, but not necessarily all, of the properties of a field. In particular, the requirements of the multiplicative operation are less strict. The most important relaxation is that nonzero elements of a ring are not required to have multiplicative inverses; but sometimes multiplication is not even required to be commutative.

粗略的說，環(huán)一種具備域的幾乎所有矗蕊，但不是所有性質(zhì)的代數(shù)結(jié)構(gòu)短蜕。尤其是，對乘法運算的要求就沒那么嚴(yán)格傻咖，最重要的放松之處是不要求環(huán)中的非零元具有乘法逆朋魔，而且有時乘法甚至不被要求是可交換的。

3 Creating New Structures Out of Old Ones （從老結(jié)構(gòu)中創(chuàng)造出新結(jié)構(gòu)）

Examples make it much easier to answer basic questions. If you have a general statement about structures of a given type and want to know whether it is true, then it is very helpful if you can test it in a wide range of particular cases. If it passes all the tests, then you have some evidence in favor of the statement. If you are lucky, you may even be able to see why it is true; alternatively, you may find that the statement is true for each example you try, but always for reasons that depend on particular features of the example you are examining. Then you will know that you should try to avoid these features if you want to find a counterexample. If you do find a counterexample, then the general statement is false, but it may still happen that a modification to the statement is true and useful. In that case, the counterexample will help you to find an appropriate modification.

有了例子卿操，回答一些基本的問題變得容易不少警检。如果我們有了一個關(guān)于某個給定類型的結(jié)構(gòu)的一般命題，而又想知道它是否正確硬纤，這時解滓，如果能夠用諸多個案來檢驗這個命題，會很有幫助筝家。如果這個命題通過了所有的檢驗洼裤，就有了有利于這個命題的證據(jù)。如果運氣好，我們也許還能看出這個命題為什么是正確的。另外开睡，也可能發(fā)現(xiàn)這個命題對于你進行檢驗的每一個例子都是對的，但是都僅僅是因為所用例子本身的特別之處移国，這個時候我們就會知道，在尋找反例時需要怎樣避免這些特別之處道伟。如果確實找到了一個反例迹缀，那么這個一般的命題當(dāng)然不成立了使碾，然而有可能這個命題在經(jīng)過某些修改以后，依然成立并且有用祝懂。在這種情況下票摇，反例就會幫助我們找到適當(dāng)?shù)男薷摹?/p>

Even though Q(i) is contained in C, it is a more interesting field in some important ways. But how can this be? Surely, one might think, an object cannot become more interesting when most of it is taken away. But a moment’s further thought shows that it certainly can: for example, the set of all prime numbers contains fascinating mysteries of a kind that one does not expect to encounter in the set of all positive integers.

......and in many other fields of a similar kind, we can ask which polynomial equations have solutions. This turns out to be a deep and important question that simply does not arise in the larger field C.

雖然 Q(i)包含在C中，但它在某些很重要的角度上是一個更有意思的域砚蓬。為什么會這樣子呢矢门？人們肯定以為如果把一個對象的絕大部分都拿走了，它不可能變得更有意思灰蛙。然而進一步想象一下祟剔，就會發(fā)現(xiàn)這確實是可能的：例如所有素數(shù)的集合會擁有某種特別迷人的、而不可能為所有正整數(shù)的集合所具備的特性摩梧。

……而且在物延，許多類似于Q(i)的域中，我們可以問哪些多項式方程有解障本。這在后來被證明是一個非常深刻而且重要的問題教届，但在更大的域C中，這樣的問題根本就不會出現(xiàn)（因為代數(shù)的基本定理告訴我們驾霜，每一個多項式方程在C內(nèi)都有解）。

We will now convert Q[x] into a field in what may at first seem a rather strange way: by regarding the polynomial

as “equivalent” to the zero polynomial. To put this another way, whenever a polynomial involves x^3 we will allow ourselves to replace x^3 by x+1, and we will regard the new polynomial that results as equivalent to the old one.

我們現(xiàn)在要用一種乍一看非常奇怪的方法买置，來把Q[x]（具有有理數(shù)系數(shù)的多項式的集合）變成一個域粪糙，方法就是，認(rèn)為

等價于零多項式忿项。換句話說蓉冈，一旦一個多項式里面有X^3的話，我們就可以把它換成x+1轩触，并且認(rèn)為這樣得出的新多項式等價于原來的多項式寞酿。

All polynomials that are not equivalent to zero (that is, are not multiples of x^3?x?1) have multiplicative inverses in this generalized sense.

所有不等價于零的多項式，都在這個廣義的意義下具有乘法逆脱柱。

We simply decide that when two polynomials are equivalent, we will regard them as equal, and we denote the resulting mathematical structure by Q[x]/(x^3 - x - 1). This structure turns out to be a field, and it turns out to be important as the smallest field that contains Q and also has a root of the polynomial x^3 - x - 1.

我們只是簡單地規(guī)定將兩個等價的多項式視為相等伐弹，并把得到的數(shù)學(xué)結(jié)構(gòu)記為Q[x]/(x^3 - x - 1)，這個結(jié)構(gòu)結(jié)果被證明是一個域榨为，而且還是個重要的域惨好，因為它是包含Q且擁有多項式x^3 - x - 1的根的最小的域。

We define two expressions ab and cd to be equivalent if ad = bc and we regard equivalent expressions as denoting the same number. Notice that the expressions can be genuinely different, but we think of them as denoting the same object. If we do this, then we must be careful whenever we define functions and binary operations.

……In general, it is essential to check that if you put equivalent objects in then you get equivalent objects out.

按：上面這里其實就是對有理數(shù)的約分的含義做了推廣随闺。

我們定義只要ad = bc那么ab和cd 這兩個表達式就等價日川，并且我們將等價的表達式看作在標(biāo)記同一個數(shù)。注意這些表達式可能的確不一樣矩乐，但我們將其視為對同一個對象的標(biāo)記龄句。如果我們這樣做，在我們定義函數(shù)和二元運算的時候就要十分小心。

一般而言分歇，最起碼要驗證透葛，如果輸入的是等價的對象，（函數(shù)或二元運算）輸出的也應(yīng)該是等價的對象卿樱。

Why is the word “quotient” used? Well, a quotient is normally what you get when you divide one number by another, so to understand the analogy let us think about dividing 21 by 3. We can think of this as dividing up twenty-one objects into sets of three objects each and asking how many sets we get.

這里為什么我們使用了“商”這個詞僚害？商通常是指當(dāng)用某個數(shù)去分割（divide，在英語里繁调，既有除的意思萨蚕，也有分割的意思）另外一個數(shù)時所得到的東西。為了理解這個比喻蹄胰，我們考慮21除以3岳遥，我們可以認(rèn)為，這是把21個對象分成了3個對象一組裕寨，然后問一共可以分得多少個組浩蓉。

...then we find that this cylinder is itself “folded around” so that if you go “upwards” by a distance of 1 then you get back to where you started. But that is what a torus is: a cylinder that is folded back into itself. (This is not the only way of defining a torus,however. For example, it can be defined as the product of two circles.)

按：上文將R^2平面上的點(x, y)和（x+1，y）定義成等價并看成相同宾袜，就會得到柱面（cylinder ）捻艳，然后又進一步將(x, y)和(x, y+1)定義成等價并看成相同，就會得到環(huán)面（torus）庆猫。

我們會發(fā)現(xiàn)认轨，這個柱面自己卷了起來，如果往上走了一段為1的距離月培，就會回到出發(fā)點嘁字。但這就是一個環(huán)面：一個被折疊成自己的柱面（然而，這不是定義環(huán)面唯一的方法杉畜，例如還可以把它定義為兩個圓周的乘積）纪蜒。

Many other important objects in modern geometry are defined using quotients. It often happens that the object one starts with is extremely big, but that at the same time the equivalence relation is very generous, in the sense that it is easy for one object to be equivalent to another. In that case the number of “genuinely distinct” objects can be quite small.

現(xiàn)代幾何中的許多重要的對象，都是用商來定義的此叠。經(jīng)常有這樣的情況纯续，一個對象很大，但同時等價關(guān)系又很寬松拌蜘，也就是一個對象杆烁，很容易就與另外一個對象等價了，在這個情況下简卧，真正不同的對象的數(shù)目可能很小兔魂。

One often starts with a hopelessly large and complicated structure but “divides out most of the mess” and ends up with a quotient object that has a structure that is simple enough to be manageable while still conveying important information. Good examples of this are the Fundamental Group [IV.10 §3] and the Homology and Cohomology Groups [IV.10 §2] of a topological space; an even better example is the notion of a Moduli Space [IV.8].

通常是從一個大的令人絕望而又極為復(fù)雜的對象出發(fā)，但將絕大部份的亂七八糟的部分都分出來除掉了（divides out）举娩，結(jié)果得到的商結(jié)構(gòu)足夠簡單析校，而且能夠處理构罗，與此同時，依舊能傳遞重要的信息智玻。基本群遂唧、拓?fù)淇臻g的同調(diào)群與上同調(diào)群都是好例子，牡跎荩空間甚至是一個更好的例子盖彭。

4 Functions between Algebraic Structures （代數(shù)結(jié)構(gòu)之間的函數(shù)）

A function that preserves structure is generally known as a homomorphism.

一個保持結(jié)構(gòu)的函數(shù)就稱為一個同態(tài)（homomorphism）。

An isomorphism between two structures X and Y is a homomorphism f : X → Y that has an inverse g : Y → X that is also a homomorphism.

兩個結(jié)構(gòu)之間的同構(gòu)（isomorphism ）就是這樣的一種同態(tài)：同態(tài)f : X → Y的逆g : Y → X也是一個同態(tài)页滚。

An isomorphism is a homomorphism that is also a Bijection [I.2 §2.2].That is, f is a one-to-one correspondence between X and Y that preserves structure.

一個同構(gòu)就是同時也是雙射的同態(tài)召边。也就是說，f是X和Y之間的一一對應(yīng)裹驰，并且保持了結(jié)構(gòu)隧熙。

雖然很基礎(chǔ)，但還是將[I.2 §2.2]中涉及雙射的段落摘錄如下：

If we want to undo the effect of a function f : A → B, then we can, as long as we avoid the problem that occurred with the approximating function discussed earlier. That is, we can do it if f(x) and f(x') are different whenever x and x' are different elements of A. If this condition holds, then f is called an injection. On the other hand, if we want to find a function g that is undone by f , then we can do so as long as we avoid the problem of the integer-doubling function. That is, we can do it if every element y of B is equal to f(x) for some element x of A (so that we have the option of setting g(y) = x). If this condition holds, then f is called a surjection. If f is both an injection and a surjection, then f is called a bijection. Bijections are precisely the functions that have inverses.

對于一個函數(shù)f : A → B幻林，如果只要當(dāng) f(x)和f(x')不同的時候贞盯，x和x'總不相同，我們就總是可以消除函數(shù)的效果（使f(x)變回x）沪饺，這時躏敢，f被稱之為一個單射（injection）。

評：單射就是（B中的元素）只要被映射過來随闽，就是（從A）唯一地（即“單”）映射過來（即“射”）父丰。

另一方面，只要B中的每一個元素y都等于A中某個元素x的f(x)掘宪，我們就總能找到一個能被f消除效果的函數(shù)g，這時f被稱為一個滿射（surjection）攘烛。

評：滿射就是（B中的元素）每個（即“滿”）都能（從A）映射過來（即“射”）魏滚。

一個既是單射又是滿射的函數(shù)f，就是一個雙射（bijection）坟漱。雙射正是那些有逆的函數(shù)鼠次。

評：滿射解決的是“有”的問題，單射解決的是“只有”的問題芋齿，所以雙射就是“有且只有”腥寇，所以B中每個元素都能找到映射的來源，而且來源還唯一觅捆，這時映射的這個唯一來源赦役，就是逆。所以“雙射”里的“雙”字更多是“成雙成對”的意思栅炒，更好的譯法或許是“對射”掂摔。費了這些口舌术羔，就是想解釋清楚這些譯法都是什么意思，當(dāng)年學(xué)的時候乙漓，挺煩這些不好記的中文譯名的级历。英文術(shù)語里，in-前綴代表“進入叭披、里內(nèi)”寥殖，sur-前綴代表“在…..之上” ，其實也不是太好理解涩蜘。

In general, if there is an isomorphism between two algebraic structures X and Y, then X and Y are said to be isomorphic (coming from the Greek words for “same” and “shape”). Loosely, the word “isomorphic” means “the same in all essential respects,” where what counts as essential is precisely the algebraic structure. What is absolutely not essential is the nature of the objects that have the structure.

一般地說嚼贡，兩個代數(shù)結(jié)構(gòu)X和Y間若有同構(gòu)的函數(shù)關(guān)系，就說X同構(gòu)于Y皱坛。同構(gòu)中的iso和morphic分別源自希臘單詞“相同”和“形狀”编曼。粗略地說，同構(gòu)這個詞的意思就是“在所有本質(zhì)的方面都相同”剩辟。算作本質(zhì)的正是代數(shù)結(jié)果掐场，而絕對不屬于本質(zhì)的，就是具有這種結(jié)構(gòu)的對象自身的本性贩猎。

An automorphism of X is a function from X to itself that preserves the structure (which now comes in the form of statements like ab = c). The composition of two automorphisms is clearly a third, and as a result the automorphisms of a structure X form a group. Although the individual automorphisms may not be of much interest, the group certainly is, as it often encapsulates what one really wants to know about a structure X that is too complicated to analyze directly.

X的自同構(gòu)是熊户，一個能夠保持結(jié)構(gòu)的、到X自身的函數(shù)吭服。兩個自同構(gòu)的復(fù)合顯然還是一個自同構(gòu)嚷堡，于是代數(shù)結(jié)構(gòu)X的所有自同構(gòu)可以形成一個群。雖然作為個體的自同構(gòu)并不那么有趣艇棕，自同構(gòu)的群蝌戒，卻很有意思。這類群往往蘊含了我們關(guān)于一個結(jié)構(gòu)真正想知道的信息沼琉，這些信息往往過于復(fù)雜北苟，無法直接分析。

So f takes every rational number to itself. What can we say about f(√ 2)? Well, f(√ 2)f (√ 2) = f(√ 2 · √ 2) = f(2) = 2, but this implies only that f(√ 2) is √ 2 or ? √ 2. It turns out that both choices are possible: one automorphism is the “trivial” one f(a + b √ 2) = a + b √ 2 and the other is the more interesting one f(a + b √ 2) = a ? b √ 2. This observation demonstrates that there is no algebraic difference between the two square roots.

f把每一個有理數(shù)都變成自身打瘪，那f(√ 2)會是多少呢友鼻？從 f(√ 2)f (√ 2) = f(√ 2 · √ 2) = f(2) = 2可知f(√ 2)是√ 2或? √ 2。究竟是哪一個闺骚？其實彩扔，兩種選擇都是可能的：一個自同構(gòu)是平凡的：f(a + b √ 2) = a + b √ 2；另外一個更為有趣：f(a + b √ 2) = a ? b √ 2僻爽。這個觀察說明了虫碉，兩個平方根并沒有代數(shù)上的區(qū)別。

The automorphism groups associated with certain field extensions are called Galois Groups [III.30], and are a vital component of the proof of the insolubility of the quintic [V.24], as well as of large parts of algebraic number theory (see Algebraic Numbers [IV.3]).

與部分域擴張相關(guān)聯(lián)的自同構(gòu)群被稱為伽羅瓦群进泼，而且是對五次方程的不可解性而言不可或缺的成分蔗衡。同時它也是代數(shù)數(shù)論相當(dāng)大一部分內(nèi)容纤虽，詳見代數(shù)數(shù)。

注：中文版有一段英文電子版中沒有的绞惦、關(guān)于同態(tài)關(guān)系中的核（kernel）的討論：核是X中所有使得f(x)為Y中的恒等元的那些x的集合逼纸，是X的有趣的子結(jié)構(gòu)；環(huán)同態(tài)的核必然是一個理想[III.81]济蝉。

Let V be another vector space of functions, and let u be a function of two variables. (The functions involved have to have certain properties for the definition to work, but let us ignore the technicalities.) Then we can define a linear map T on the space V by the formula

.
Definitions like this one can be hard to take in, because they involve holding in one’s mind three different levels of complexity. At the bottom we have real numbers, denoted by x and y. In the middle are functions like f , u, and Tf, which turn real numbers (or pairs of them) into real numbers. At the top is another function, T, but the “objects” that it transforms are themselves functions: it turns a function like f into a different function Tf. This is just one example where it is important to think of a function as a single, elementary “thing” rather than as a process of transformation. (See the discussion of functions in the language and grammar of mathematics [I.2 §2.2].)
Another remark that may help to clarify the definition is that there is a very close analogy between the role of the two-variable function u(x,y) and the role of a matrix aij (which can itself be thought of as a function of the two integer variables i and j). Functions like u are sometimes called kernels.
For more about linear maps between infinite-dimensional spaces, see Operator Algebras [IV.19] and Linear Operators [III.52].

第喳。

像這樣的定義可能難以接受，因為它們涉及到三個層次的復(fù)雜性踱稍。在底層有兩個實數(shù)曲饱，可以表示為x和y。中間一層有一些函數(shù)珠月，如f扩淀、u和Tf，它們都是將實數(shù)（或?qū)崝?shù)對）映射為實數(shù)啤挎。最頂層是另外一個函數(shù)T驻谆，但它所轉(zhuǎn)換映射的對象本身就是函數(shù)：它將一個函數(shù)f變成另外一個函數(shù)Tf。這個例子說明了如下思維方式的重要性：將函數(shù)看作單一和基礎(chǔ)的東西而非一個轉(zhuǎn)換的過程庆聘。（參見[I.2 §2.2]中對此的討論）另外一個有助于理清這個定義的點是：二元函數(shù)u(x,y)的角色與矩陣a_ij極其類似胜臊。（矩陣a_ij自己也可以被看作兩個整數(shù)變量i和j的函數(shù)）

關(guān)于無限空間之間的線性映射，可以參考算子代數(shù)和線性算子

In many cases the eigenvectors and eigenvalues associated with a linear map contain all the information one needs about the map, and in a very convenient form. Another answer is that linear maps occur in many different contexts, and questions that arise in those contexts often turn out to be questions about eigenvectors and eigenvalues

在許多情況下伙判，線性映射的本征向量與本征值区端，包含了關(guān)于這個線性映射我們所有需要了解的信息，而且是以非常方便的形式澳腹。線性映射出現(xiàn)在很多情境中，這些情境中出現(xiàn)的問題往往正是關(guān)于本征向量和本征值的問題杨何。

function [III.25] e^x: that its derivative is the same function. In other words, if f(x) = e^x, then f '(x) = f(x). Now differentiation, as we saw earlier, can be thought of as a linear map, and if f '(x) = f(x) then this map leaves the function f unchanged, which says that f is an eigenvector with eigenvalue 1. More generally, if g(x) = e^(λx), then g'(x) = λ e^(λx) = λg(x), so g is an eigenvector of the differentiation map, with eigenvalue λ. Many linear differential equations can be thought of as asking for eigenvectors of linear maps defined using differentiation.

指數(shù)函數(shù) e^x 的導(dǎo)數(shù)是其自身酱塔。換句話說，如果f(x)=e^x危虱，那么f'(x)=f(x)羊娃。這樣微分運算就可以被看作一種線性映射。如果f '(x) = f(x)埃跷，那么這個映射使函數(shù)f保持不變蕊玷，這說明f是一個具備本征值1的本征向量邮利。更一般的，如果g(x) = e^(λx)垃帅，那么g'(x) = λ e^(λx) = λg(x)延届，這樣g就是微分映射的一個本征向量，其本征值為λ贸诚。許多線性微分方程可以被視為在求用微分運算定義的線性映射的本征向量方庭。