# Irreducibility and Roots in ##\Bbb{C}##

• Bashyboy
In summary, the theorem states that if ##p(x) \in \Bbb{Q}[x]## is an irreducible polynomial, then ##p(x)## has no repeated roots in ##\Bbb{C}##. The proof of this statement involves proving the contrapositive, which states that if ##p(x)## has a repeated root in ##\Bbb{C}##, then ##p(x)## is not irreducible. This is done by showing that in this case, the polynomial ##p(x)## can be factored into two polynomials of smaller degree, thus contradicting its irreducibility.
Bashyboy

## Homework Statement

Prove that if ##p(x) \in \Bbb{Q}[x]## is an irreducible polynomial, then ##p(x)## has no repeated roots in ##\Bbb{C}##.

## Homework Equations

I will appeal to the theorem I attempted to prove here: https://www.physicsforums.com/threads/repeated-roots-and-being-relatively-prime-w-derivative.927168/

Let ##k## be a subfield of the field ##K##. If ##f(x), g(x) \in k[x]##, then their gcd in ##k[x]## is equal to their gcd in ##K[x]##.

If ##k## is a field, then a polynomial ##p(x) \in k[x]## is irreducible if and only if ##\deg (p) = n \ge 1##and there is no factorization in ##k[x]## of the form ##p(x) = g(x) h(x)## in which both factors have a degree smaller than ##n##.

## The Attempt at a Solution

I will prove the contrapositive. Suppose that ##p(x)## has a repeated root in ##\Bbb{C}##. Then ##(p,p')_{\Bbb{Q}} = (p,p')_{\Bbb{C}} \neq 1##, and according to the theorem given in the link above, ##p## must have a repeated root in ##\Bbb{Q}##. This means that ##(x-a)^2 |p(x)##, where ##a \in \Bbb{Q}## is the repeated root of ##p##, and so ##p(x) = (x-a)^2 f(x)##. This means ##\deg(p(x)) = \deg((x-a)^2 f(x)) = 2 + \deg(f)## and hence ##\deg(f) < \deg (p)## and ##2 \le \deg (p)##. If ##2 = \deg(p)##, then ##\deg (f) = 0## and so ##f(x) = r \in \Bbb{Q}##, which gives us ##p(x) = r(x-a)^2 = r(x-a)(x-a)##, each factor having a degree smaller than ##p##'s. Thus ##p(x)## is not irreducible whether ##\deg(p) = 2## or ##\deg (p) < 2##.

How does this sound?

Correct. The proof of the statement you used is the interesting part.

Bashyboy

## 1. What is meant by "irreducibility" in ##\Bbb{C}##?

Irreducibility in ##\Bbb{C}## refers to a polynomial function that cannot be factored into smaller polynomials over the field of complex numbers. This means that it does not have any roots or solutions in ##\Bbb{C}##, making it a fundamental and indivisible polynomial.

## 2. What is the difference between irreducibility and primality in ##\Bbb{C}##?

The terms irreducibility and primality are often used interchangeably in ##\Bbb{C}##, but they have distinct meanings. An irreducible polynomial cannot be factored into smaller polynomials over ##\Bbb{C}##, while a prime polynomial cannot be factored into smaller polynomials over any field, including ##\Bbb{C}##. Essentially, all prime polynomials are irreducible, but not all irreducible polynomials are prime.

## 3. How can I find the roots of an irreducible polynomial in ##\Bbb{C}##?

Finding the roots of an irreducible polynomial in ##\Bbb{C}## can be challenging and may require advanced mathematical techniques. One approach is to use the Fundamental Theorem of Algebra, which states that every polynomial of degree ##n## has exactly ##n## complex roots (including repeats). Another approach is to use the Rational Root Theorem to narrow down the possible roots and then use synthetic division to find the actual roots.

## 4. Can an irreducible polynomial have complex roots?

Yes, an irreducible polynomial can have complex roots. In fact, if the degree of the polynomial is greater than 2, it is almost certain to have complex roots. This is because the Fundamental Theorem of Algebra states that a polynomial of degree ##n## has exactly ##n## complex roots. Therefore, an irreducible polynomial of degree greater than 2 must have complex roots.

## 5. Are there any applications of irreducible polynomials in ##\Bbb{C}##?

Yes, irreducible polynomials have a wide range of applications in mathematics, physics, and engineering. In mathematics, they are used to study the structure of finite fields and to construct error-correcting codes. In physics, they are used to model and analyze complex systems, such as chaotic behavior in fluid dynamics. In engineering, they are used in signal processing and control theory to design stable systems.

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