Name: 積分域(抽象代數) (Integral Domains (Abstract Algebra))
Uploaded: 2021-01-14T08:51:12.000Z
Duration: 7 min 34 s
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Suppose you want to solve the equation X² - 4 = 0.

In regular algebra you would factor the left-hand side giving you (x-2)(x+2)=0.

情態助動詞 B1

Here the product of two terms is zero, so one of the terms must be zero.

So this equation has two solutions: 2 and -2.

But to solve this equation we used a BIG assumption:

that if the product of two terms is zero,  then one of the terms must be zero.

In Abstract Algebra, this isn’t always true.

Let’s see an example where this technique of solving equations does NOT work.

We’ll solve the equation x² + 5x + 6 = 0 in the integers mod 12.

If you factor the left-hand side, you get (x+2)*(x+3) is congruent to 0 mod 12.

If you set each term to 0 mod 12,  you get x=10 or x=9.

Since there are only 12 numbers in the ring of integers mod 12,

let’s go ahead and check all possibilities.

We’ll plug in 0 through 11 into the polynomial x² + 5x + 6 and see what we get.

If you plug in 0 you get 6, so x=0 is not a solution.

If you plug in 1 you get 12 which is congruent to 0 mod 12, so 1 IS a solution.

Here’s a solution to the equation that we did NOT find by factoring.

And let’s quickly fill out the table for the remaining numbers.

We see this equation has not two, but FOUR solutions: 1, 6, 9 and 10.

Factoring only identified the solutions 9 and 10.

To see the problem, let’s take a closer look at the equation.

After factoring, we have (x+2)*(x+3) is congruent to 0 mod 12.

Setting each term to zero gave us two of the solutions: 9 and 10.

To see why, look what happens when you plug in 1.

This gives you 3 × 4, which is 0 mod 12.

And if you plug in 6, you get 8 × 9 which is also 0 mod 12.

Working mod 12, it’s possible to multiply two non-zero numbers together and get 0.

We say the integers mod 12 have “zero divisors,”

and it’s the zero divisors which make solving equations more difficult.

With real and complex numbers, this doesn’t happen.

That’s why in regular algebra, you learn to solve many equations by factoring

But in abstract algebra, this technique is not good enough.

I’d like to take a moment and talk about the term “zero divisors.”

This name was chosen because division can be defined in terms of multiplication.

For example, when working with integers,  we say B divides A

if  B × C = A for some integer C. From this definition, the term “zero divisors” makes sense.

In the integers mod 12, 3 × 4 = 0, so both 3 and 4 divide 0.

Let’s return to the equation X² + 5X + 6 = 0,

except this time we’ll work mod 11 instead of mod 12.

The integers mod 11 do NOT have any zero divisors.

The only way the product of two numbers  is a multiple of 11

is if one of the numbers is divisible by 11.

And none of the integers 1 through 10 is a multiple of 11,

so this ring does not have any zero divisors.

Like before, we can factor the left-hand side giving us

(X + 2)*(X + 3) is congruent to 0 mod 11.

Setting each term to zero gives us x = 9 and x = 8.

Let’s see if there are any other solutions by plugging in all the integers mod 11.

Let’s go ahead and fill out the rest of the table.

We see that the only solutions are 8 and 9, the two solutions we found by factoring.

When a ring R does NOT have any zero divisors, the traditional technique of solving an equation

by factoring and setting the terms to zero DOES work.

For this reason, there’s a term for such rings: integral domains.

Here’s the complete definition: an integral domain is a commutative ring R

with a multiplicative identity 1 and NO zero divisors.

A natural question is why do we require an integral domain to be commutative?

After all, it seems that the most important thing, at least for solving equations,

One reason, is when the idea of an integral domain was developed, much of the focus was

on generalizing the integers for use in number theory.

Hence the word “integral” in “integral domain.”

There is, however, a term for an arbitrary ring with no zero divisors: a DOMAIN.

But you’ll encounter integral domains more frequently.

Integral domains have another useful property: the cancellation property.

Suppose you want to find the solutions to the equation 2X = 6Y.

Your instinct might be to first simplify this by dividing both sides by 2,

But when working with a ring, you may not be able to divide.

So a different approach would be to factor both sides, then cancel the 2s,

Factoring and cancelling does not require division.

Look at the equation 3x = 6y in the ring of integers mod 12.

This equation has multiple solutions, but one solution is x = 6 and y = 1.

This is because 18 is congruent to 6 mod 12.

But look what happens if you factor out 3 then cancel.

Now if you plug in x=6 and y=1 you get  6 is congruent to 2 mod 12, which is false.

So in the original equation, 6 and 1 is a solution, but after cancelling, it is not.

To find out, suppose a*x = a*y  in some ring R, and 'a' is not 0.

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積分域(抽象代數) (Integral Domains (Abstract Algebra))

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