Application of cyclotomic polynomial

• math_grl
In summary: Q}[x]/(p), which is equivalent to saying that they are equal in \mathbb{Q}[x]. Therefore, we can rewrite the right hand side as \frac{(p-1)!}{(k-1)!(p-k)!} + \frac{(p-2)!}{(k-2)!(p-k-1)!} + \cdots + \frac{(p-k)!}{0!(p-k)!} = \frac{p!}{k!(p-k)!} = {p \choose k}.In summary, we have shown that the expression {p \choose k} is equal to \sum^{k+1}_{i=1} {p-i \choose
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Homework Statement

Show that $${p \choose k} = \sum^{k+1}_{i=1} {p-i \choose p-k-1}$$ where $$\forall k < p \in \mathbb{Z}$$ and $$p$$ a prime.

Homework Equations

This is part (b) to a problem. Part (a) is showing that $$1 + x + x^2 + \cdots + x^{p-1}$$ is irreducible in $$\mathbb{Q}[x]$$.

The Attempt at a Solution

I don't know how to apply part (a) to part (b)...Worked out the equality algebraically, then could probably do it by finite induction but think there has to be a shorter easier way that's supposed to follow from the previous part. Help.

Thank you for your post. It seems like you are working on a problem involving binomial coefficients and their relationship to irreducible polynomials. In order to solve this problem, we can use the fact that the polynomial 1 + x + x^2 + \cdots + x^{p-1} is irreducible in \mathbb{Q}[x]. Since this is part (a) of the problem, we can assume that this fact has already been proven.

Now, let's take a closer look at the expression {p \choose k}. By definition, this is the number of ways to choose k elements from a set of p elements. We can rewrite this as {p \choose k} = \frac{p!}{k!(p-k)!}. This is where the irreducibility of 1 + x + x^2 + \cdots + x^{p-1} comes into play.

Since this polynomial is irreducible, it has no roots in \mathbb{Q}. Therefore, it also has no roots in \mathbb{Q}[x]/(p), the field of p-adic numbers. This means that we can use this field to "count" the number of ways to choose k elements from a set of p elements. In this field, we can think of the polynomial 1 + x + x^2 + \cdots + x^{p-1} as representing the set of all possible k-element subsets of a p-element set.

Now, let's focus on the right hand side of the expression we are trying to prove, \sum^{k+1}_{i=1} {p-i \choose p-k-1}. This can be rewritten as \frac{(p-1)!}{(k-1)!(p-k)!} + \frac{(p-2)!}{(k-2)!(p-k-1)!} + \cdots + \frac{(p-k)!}{0!(p-k)!}. Notice that this is exactly the same as the expression for {p \choose k}, except with the terms in the denominator switched around.

This is where the fact that p is a prime number comes into play. Since p is prime, we know that p-i and p-k-1 are relatively prime for all i < k. This means that the fractions in the right hand side of the expression can be reduced to their simplest form in \

1. What is a cyclotomic polynomial?

A cyclotomic polynomial is a type of polynomial that is defined by the roots of unity, which are complex numbers that have a value of 1 when raised to a certain power. These polynomials have important applications in number theory and algebraic geometry.

2. How are cyclotomic polynomials used in mathematics?

Cyclotomic polynomials have various applications in mathematics, including in the study of algebraic number fields, Galois theory, and the construction of finite fields. They also have connections to other areas of mathematics, such as topology and combinatorics.

3. What is the significance of the name "cyclotomic polynomial"?

The term "cyclotomic" comes from the Greek word "kyklos" meaning circle, as these polynomials are closely related to the roots of unity, which form a circle on the complex plane. Additionally, the word "cyclo" refers to the fact that these polynomials are closely connected to cyclic groups in abstract algebra.

4. How are cyclotomic polynomials computed?

There are various methods for computing cyclotomic polynomials, such as using the factorization of the polynomial into irreducible factors or using recursion formulas. In some cases, they can also be computed using trigonometric identities.

5. What are some real-world applications of cyclotomic polynomials?

Cyclotomic polynomials have applications in cryptography, specifically in the construction of secure hash functions and in the design of certain encryption algorithms. They are also used in error-correcting codes and in digital signal processing.

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