Searching for Closure: Mult. Inverses & Addition

In summary, the conversation discusses presenting closure under multiplication inverses and closure under addition, with the main focus being on how to prove that (a+b)^p=a^p+b^p in a field of characteristic p. The use of the binomial theorem and the characteristic of the field are key in these proofs.
  • #1
stripes
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Homework Statement



attachment.php?attachmentid=63040&stc=1&d=1382007608.jpg


Homework Equations





The Attempt at a Solution



Well thankfully I just have to present closure under mult. inverses and closure under addition. But I seem to be going in circles...if a is in G, then we need to show that a-1 is also in G.

So a*a-1 = 1F, but is a-1 in G...so we can write a = ap, then if we can write a-1 = (ap)-1, then we'll be good. But how the heck do I do that?

For closure under addition, I'm not really sure how to use the binomial theorem here, since we know that a + b means we can write them both as ap + bp, and now we've got to show that ap + bp can be written as (α + β)p. Not sure how to get this one started either.

Any help is appreciated, thanks in advance.
 

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  • #2
stripes said:

Homework Statement



attachment.php?attachmentid=63040&stc=1&d=1382007608.jpg


Homework Equations





The Attempt at a Solution



Well thankfully I just have to present closure under mult. inverses and closure under addition. But I seem to be going in circles...if a is in G, then we need to show that a-1 is also in G.

So a*a-1 = 1F, but is a-1 in G...so we can write a = ap, then if we can write a-1 = (ap)-1, then we'll be good. But how the heck do I do that?

For closure under addition, I'm not really sure how to use the binomial theorem here, since we know that a + b means we can write them both as ap + bp, and now we've got to show that ap + bp can be written as (α + β)p. Not sure how to get this one started either.

Any help is appreciated, thanks in advance.

I'm not sure why you are stumbling around with the multiplicative proof. If a=a^p then a^(-1)=(a^p)^(-1). But that's the same thing as (a^(-1))^p isn't it? And for addition, you use the binomial theorem to prove that (a+b)^p=a^p+b^p in F. This is where the field having characteristic p is really important.
 

1. What is the purpose of searching for closure in mathematics?

The concept of closure refers to the idea that an operation performed on elements from a certain set will result in another element from the same set. In mathematics, searching for closure helps us determine whether a set of numbers is closed under a given operation, which is important for understanding the behavior and properties of that operation.

2. How is closure related to the concept of multiplicative inverses?

In order for a set to be closed under multiplication, every element in that set must have a multiplicative inverse, meaning that when multiplied together, they will result in the identity element of that set (usually 1). Therefore, searching for closure involves determining whether a set has all the necessary multiplicative inverses for a given operation.

3. Can a set be closed under addition but not under multiplication?

Yes, it is possible for a set to be closed under addition but not under multiplication. For example, the set of integers is closed under addition, as adding any two integers will always result in another integer. However, it is not closed under multiplication, as multiplying certain integers may result in a decimal or fraction, which is not an integer.

4. How does the concept of closure extend to other mathematical operations?

Closure extends to all mathematical operations, not just multiplication and addition. A set can also be closed under subtraction, division, and more complex operations such as composition of functions. In each case, the operation must be performed on elements from the set and result in another element from the same set in order for the set to be considered closed.

5. Why is the concept of closure important in real-world applications?

Closure has many real-world applications, especially in fields such as computer science, physics, and economics. For example, in computer algorithms, closure is used to ensure that a program will terminate and provide a correct result. In physics, closure is important for understanding the behavior of systems and predicting outcomes. In economics, closure is used to analyze the stability of economic systems and determine the impact of certain actions.

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