 ## 字幕列表 影片播放

• 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

• 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.

• 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.

• Thank you for watching Socratica. Were a small team, making beautiful, high-quality,

• CHALLENGING math and science videos. We put our heart and soul and ALL of our own money

• into making these videos for you. I’ll be honest with you - it’s not easy sometimes.

• We’d love to make more videos, faster, but our resources are limited. If you find our

• videos helpful - if you value what we do, please consider becoming our Patron on Patreon.

• Even the smallest donation goes a long way. Thank you.

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

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

• 1 0
林宜悉 發佈於 2021 年 01 月 14 日