Field question Divisibility

In summary, If g(x)|f(x) and f(x)|g(x), then f(x)=kg(x) for some k in F. This is shown by using the fact that F[x] is an integral domain and the degrees of s(x) and r(x) must be 0, resulting in f(x)=kg(x).
  • #1
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Let f(x), g(x) be in F[x]. Show that if g(x)|f(x) and f(x)|g(x), then f(x)=kg(x) for some k in F.

Since g(x)|f(x), then f(x)=g(x)r(x) for some r(x) in F[x].
Similarily, since f(x)|g(x), then g(x)=f(x)s(x)
So f(x)=f(x)s(x)r(x)
 
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  • #2
I tried doing something with degrees.
deg(fsr)=deg(f)+deg(s)+deg(r)
So deg(f)=deg(f)+deg(s)+deg(r)
0=deg(s)+deg(r)
deg(s)=-deg(r)
 
  • #3
What do you mean with F(x)? You mean polynomials or rational functions? The standard notation for polynomials is F[x]...

Now, you got that f(x)=f(x)s(x)r(x). Now apply that F[x] is an integral domain...
 
  • #4
I mean polynomials F[x]
 
  • #5
I haven't learned about integral domains yet
 
  • #6
An integral domain is just a ring such that ab=0 implies a=0 and b=0.
Now, since F[x] is clearly an integral domain, what does this imply for the equation f(x)=f(x)s(x)r(x)??
 
  • #7
kathrynag said:
I tried doing something with degrees.
deg(fsr)=deg(f)+deg(s)+deg(r)
So deg(f)=deg(f)+deg(s)+deg(r)
0=deg(s)+deg(r)
deg(s)=-deg(r)

you almost got it, you know that deg(s)>=0 and deg(r)>=0 and deg(s)=-deg(r) hence it should be deg(s)=deg(r)=0.
 
  • #8
Guess I don't know what to do with that that information
 
  • #9
I don't how to go from there to f(x)=kg(x)
 
  • #10
er deg(s)=0 means there is a constant c in F such that s(x)=c.
 
  • #11
ok so s(x)=c
g(x)=cf(x)
but we had r(x)=s(x)=0 so there is r(x)=k
f(x)=kg(x)
 

1. What is divisibility?

Divisibility is the mathematical concept of determining whether one number is evenly divisible by another number without leaving a remainder.

2. How do you test for divisibility?

There are different rules for testing divisibility depending on the divisor. Some common rules include checking if the number is even, if the sum of its digits is divisible by 3, or if the last two digits form a number divisible by 4 or 8.

3. What is the significance of divisibility in mathematics?

Divisibility is important in mathematics because it allows us to simplify and solve problems more easily. It is also a fundamental concept in number theory and is used in many other areas of mathematics, such as algebra and geometry.

4. Can any number be divided by 0?

No, division by 0 is undefined in mathematics. It is not possible to divide any number by 0 and get a meaningful answer.

5. How is divisibility related to prime numbers?

Prime numbers are only divisible by 1 and themselves. This property makes them important in determining the factors of a number. If a number is only divisible by 1 and itself, it is a prime number. Divisibility rules can also help determine whether a number is prime or not.

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