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  • In abstract algebra, there is a wide variety of operations: geometric transformations,

  • function composition, matrix multiplicationBut many sets of elements have the familiar

  • operations from arithmetic: addition, subtraction, multiplication and division. Loosely speaking,

  • if you can add and subtract, you have a group. If you can add, subtract and multiply, you

  • have a ring. But if youre lucky and get all four operations, the result is an object

  • that behaves similarly to the numbers you learned about in arithmetic and algebra. We

  • call these fields.

  • One thing weve talked about before, but it bears repeating, is that in abstract algebra,

  • subtraction is actually adding with negatives. For example, “3 minus 5” is the same as

  • 3 plusnegative 5.” So instead of sayingyou can add and subtract”, we saythere’s

  • addition and additive inverses.” Notice that wordadditive?” That’s because

  • there is more than one kind of inverse.

  • For example, “3 DIVIDED by 5” is the same as 3 times 1/5.

  • So instead of sayingyou can multiply and divide”, in abstract algebra we saythere’s

  • multiplication and multiplicative inverses.” The additive inverse of 5 is negative 5.

  • The multiplicative inverse of 5 is 1/5.

  • So in arithmetic, you learn about addition, subtraction, multiplication and division.

  • But in abstract algebra, you speak of addition, additive inverses, multiplication, and multiplicative

  • inverses. It’s a shift in thinking, but it’s key to understanding the more abstract

  • objects.

  • Were now ready to talk about fields. To motivate the definition, well start with

  • a collection of 6 groups, some of which come with additional features. By adding features

  • were familiar with from the real and complex numbers, well arrive at the full definition

  • of a field.

  • Consider these 6 sets: The integers

  • The 2-by-3 real matrices The 2-by-2 real matrices

  • The rational numbers The integers mod 5, and

  • The integers mod 6. Notice that all 6 sets are groups under addition:

  • Theyre closed under addition: you can add two elements together and the sum is in the

  • set. The negative of each element is in the set.

  • There’s an additive identity, and the associative property holds.

  • Better still, all 6 groups are COMMUTATIVE under addition.

  • So as a first pass, all 6 objects are commutative groups under addition.

  • The next feature we’d like to include is multiplication.

  • You can multiply any two integers or rational numbers together.

  • You can also multiply any two numbers mod N for any N.

  • That leaves the two sets of matrices. You can multiply two square matrices, but

  • you cannot multiply 2-by-3 matrices by each other.

  • Their dimensions are incompatible for multiplication. So only 5 of the 6 sets advance to the next

  • round of commutative groups with multiplication.

  • Just as all the groups are commutative under addition, in a field, we’d like multiplication

  • to be commutative as well. After all, the real and complex numbers are

  • both commutative, and they are a pleasure to work with.

  • The integers and rational numbers are both commutative under multiplication, so they

  • advance. Also, the integers mod N are commutative under

  • multiplication for any N. But were about to lose another candidate.

  • The 2-by-2 matrices are NOT commutative under multiplication.

  • There are an infinite number of examples where matrix multiplication is not commutative.

  • Here’s one example... So only 4 of the 6 are commutative under multiplication.

  • Next, we’d like each number to have a multiplicative inverse.

  • The additive inverse 0 is the big exception here.

  • You cannot divide by 0, so this number cannot have a multiplicative inverse.

  • But we’d like every NON-ZERO number to have a multiplicative inverse.

  • Sadly, were about to lose two more sets. In the set of integers, only 1 and -1 have

  • multiplicative inverses. None of the other integers have one.

  • For example, the inverse of 2 under multiplication is ½, which is not an integer.

  • And we lose the integers mod 6 as well. 2, 3 and 4 do not have inverses mod 6.

  • Mod 5, however, is different. Here, every non-zero number has a multiplicative

  • inverse. You can check this by looking at the multiplication

  • table for this set. So the only two sets to advance are the rational

  • numbers and the integers mod 5. By the way, these two sets both have a multiplicative

  • identity 1. This is not a surprise, since the product

  • of a number and its multiplicative inverse is 1...

  • The race is over, and we have two winners. The rational numbers and the integers mod

  • 5. These both share a similar set of properties. They are both commutative groups under addition.

  • They both have a second operation - multiplication, which makes them rings. Furthermore, multiplication

  • is commutative, so they are commutative rings. Better still, other than zero, every number

  • has a multiplicative inverse. It’s this last property which puts them over the top

  • and turns them into fields. Were now ready for the textbook definition of a field.

  • A field is a set of elements “F” with two operations: addition and multiplication.

  • Under addition, the elements are a commutative group. Under multiplication, the non-zero

  • elements are a commutative group. And addition and multiplication are linked by the distributive

  • property. This is the compact definition of a field. If you wanted, you could define a

  • field and make no mention of groups whatsoever. You could just give a complete list of all

  • the properties a field must satisfy. This is fine, but you do lose sight of the fact

  • that a field is actually two groups with two operations at the same time.

  • Let’s return to the two examples of a field we just saw: the rational numbers and the

  • integers mod 5. The rational numbers are denoted by “Q”

  • forquotient”, since every number in this field is the quotient of two integers.

  • The rationals are an infinite field, while the integers mod 5 are a finite field.

  • But the integers mod 5 are not the only finite field.

  • In fact, the integers mod “P” for ANY prime number “P” is also a field.

  • Together, these form the starting points for ALL fields.

  • That is, if you pick ANY field “F”, then it will contain one and only one of these

  • fields as a subfield. We call these fieldsprime fields”, and

  • say that “F” is an extension field. If “F” is an extension of the integers

  • mod 2, we say it hascharacteristic 2.” If it’s an extension of the integers mod

  • 3, it hascharacteristic 3.” And if it’s an extension of the integers

  • mod P, we say “F” hascharacteristic P.”

  • But if F is an extension of the rational numbers, we say it hascharacteristic 0.”

  • So the characteristic of a field tells us which prime field it extends.

  • The are an infinite number of fields in mathematics. We begin by learning about thebig 3”:

  • the rational numbers, the real numbers and the complex numbers. But there’s an infinite

  • number of infinite fields, and even an infinite number of FINITE fields! From finite fields

  • to Galois extension fields, youll find many uses for this structure.

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In abstract algebra, there is a wide variety of operations: geometric transformations,

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B2 中高級

場定義(擴展) - 抽象代數 (Field Definition (expanded) - Abstract Algebra)

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    林宜悉 發佈於 2021 年 01 月 14 日
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