Proving f(x) Divides g(x) iff g(x) in <f(x)>

  • Thread starter tinynerdi
  • Start date
In summary, we are asked to show that in a field F, if f(x) divides g(x), then g(x) belongs to the ideal generated by f(x). This can be proven by considering the definition of divisibility and the properties of prime ideals. The proof can also be easily reversed to show that if g(x) belongs to the ideal generated by f(x), then f(x) divides g(x).
  • #1
tinynerdi
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Homework Statement


let F be a field and f(x),g(x) in F[x]. Show that f(x) divides g(x) if and only if g(x) in <f(x)>


Homework Equations


let E be the field F[x]/<f(x)>


The Attempt at a Solution


<=> if f(x) divides g(x) then g(x) in <f(x)>
Proof: Suppose f(x) divides g(x)q(x). then g(x)q(x) in <f(x)>. which is maximal. Therefore <f(x)> is a prime ideal. Hence g(x)q(x) in <f(x)>. implies that either g(x) in <f(x)> giving f(x) divides g(x) or that q(x) in <f(x)> giving f(x) divides q(x). But we want that g(x) in <f(x)> giving f(x) divides g(x).

can this prove go both way if it is right?
 
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  • #2
I think this is even simpler than you think. The condition for g(x) to divide f(x) is that there is q(x) in F[x] such that f(x)=q(x)g(x), and this is exactly the condition for g(x) to belong to the ideal generated by f(x).
 

1. How do you prove that f(x) divides g(x) if and only if g(x) is in ?

To prove this statement, we need to show that both conditions are true. First, we must show that if f(x) divides g(x), then g(x) is in . This can be done by using the definition of division, which states that if f(x) divides g(x), then there exists a number k such that g(x) = kf(x). This means that g(x) is a multiple of f(x), and therefore is in . Conversely, if g(x) is in , then it can be expressed as kf(x) for some number k, proving that f(x) divides g(x).

2. What is the significance of in this statement?

represents the set of all multiples of f(x). In other words, it is the set of all possible values that can be obtained by multiplying f(x) by any integer. In this context, it is used to represent the range of possible values that g(x) can take if it is divisible by f(x).

3. Can this statement be applied to any two functions, or only specific types?

This statement can be applied to any two functions, as long as they are defined over the same domain. However, the concept of divisibility may not always make sense for certain types of functions, such as trigonometric or exponential functions.

4. How does this statement relate to the concept of greatest common divisor (GCD)?

This statement is closely related to the concept of GCD, as it essentially states that the GCD of f(x) and g(x) is equal to f(x) if and only if f(x) divides g(x). In other words, if f(x) is the largest common factor of f(x) and g(x), then it must also be a factor of g(x).

5. Is this statement always true, or are there exceptions?

This statement is always true as long as the functions involved are defined over the same domain and the concept of divisibility makes sense for those functions. However, it is important to note that this statement only considers divisibility in terms of integers and may not hold true for other number systems, such as rational or complex numbers.

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