Irreducibility

Polynomial \( P(x) \) with integer coefficients is said to be irreducible over \( \mathbb{Z}[x] \) if it cannot be written as a product of two nonconstant polynomials with integer coefficients.

Example
 

Every quadratic or cubic polynomial with no rational roots is irreducible over \( \mathbb{Z} \). Such are e.g. \( x^2-x-1 \) and \( 2x^3-4x+1 \).

One analogously defines (ir)reducibility over the sets of polynomials with e.g. rational, real or complex coefficients. However, of the mentioned, only reducibility over \( \mathbb{Z}[x] \) is of interest. Gauss’ Lemma below claims that the reducibility over \( \mathbb{Q}[x] \) is equivalent to the reducibility over \( \mathbb{Z}[x] \). In addition, we have already shown that a real polynomial is always reducible into linear and quadratic factors over \( \mathbb{R}[x] \), while a complex polynomial is always reducible into linear factors over \( \mathbb{C}[x] \).

Theorem 4.1 (Gauss’ Lemma)
 

If a polynomial \( P(x) \) with integer coefficients is reducible over \( \mathbb{Q}[x] \), then it is reducible over \( \mathbb{Z}[x] \), also.

From now on, unless otherwise specified, by irreducibility we mean irreducibility over \( \mathbb{Z}[x] \).

Problem 9
 

If \( a_1 \), \( a_2 \), \( \dots \), \( a_n \) are integers, prove that the polynomial \( P(x)=(x-a_1)(x-a_2)\cdots(x-a_n)-1 \) is irreducible.

Theorem 4.2 (Extended Eisenstein’s Criterion)
 

Let \( P(x)= a_nx^n+\dots+a_1x+a_0 \) be a polynomial with integer coefficients. If there exist a prime number \( p \) and an integer \( k\in\{0,1,\dots, n-1\} \) such that \[ p\mid a_0,a_1,\dots,a_k,\;\;p\nmid a_{k+1}\;\;\mbox{and}\;\;p^2 \nmid a_0,\] then \( P(x) \) has an irreducible factor of a degree greater than \( k \).

In particular, if \( p \) can be taken so that \( k=n-1 \), then \( P(x) \) is irreducible.

Problem 10 (IMO93.1)
 

Given an integer \( n> 1 \), consider the polynomial \( f(x)=x^n+5x^{n-1} +3 \). Prove that there are no nonconstant polynomials \( g(x),h(x) \) with integer coefficients such that \( f(x)=g(x)h(x) \).

Problem 11
 

If \( p \) is a prime number, prove that the polynomial \( \Phi_p(x)=x^{p-1}+\cdots+x+1 \) is irreducible.

In investigating reducibility of a polynomial, it can be useful to investigate its zeros and their modules. The following problems provide us an illustration.

Problem 12
 

Prove that the polynomial \( P(x)=x^n+4 \) is irreducible over \( \mathbb{Z}[x] \) if and only if \( n \) is a multiple of 4.

If the zeros cannot be exactly determined, one should find a good enough bound. Estimating complex zeros of a polynomial is not always simple. Our main tool is the triangle inequality for complex numbers: \[ |x|-|y|\leq|x+y|\leq|x|+|y|.\]

Consider a polynomial \( P(x)=a_nx^n+a_{n-k}x{n-k}+\cdots+a_1x+a_0 \) with complex coefficients (\( a_n\neq0 \)). Let \( \alpha \) be its zero. If \( M \) is a real number such that \( |a_i|< M|a_n| \) for all \( i \), it holds that \[ 0=|P(\alpha)|\geq|a_n||\alpha|^n-M|a_n| (|\alpha|^{n-k}+\cdots+|\alpha|+1)> |a_n||\alpha|^n \left(1-\frac M{|\alpha|^{k-1}(|\alpha|-1)}\right),\] which yields \( |\alpha|^{k-1}(|\alpha|-1)< M \). We thus come to the following estimate:

Theorem 4.3
 

Let \( P(x)=a_nx^n+\cdots+a_0 \) be a complex polynomial with \( a_n\neq0 \) and \( \displaystyle M=\max_{0\leq k< n}\left|\frac{a_k}{a_n} \right| \). If \( a_{n-1}=\cdots=a_{n-k+1}=0 \), then all roots of the polynomial \( P \) are less than \( 1+\sqrt[k]M \) in modulus.

In particular, for \( k=1 \), each zero of \( P(x) \) is of modulus less than \( M+1 \).

Problem 13 (Balkan Mathematical Olympiad 1989)
 

If \( \overline{a_n\dots a_1a_0} \) is a decimal representation of a prime number and \( a_n> 1 \), prove that the polynomial \( P(x)=a_nx^n+ \cdots+a_1x+a_0 \) is irreducible.

Problem 14
 

Let \( p> 2 \) be a prime number and \( P(x)=x^p-x+p \).

  • (a) Prove that all zeros of polynomial \( P \) are less than \( \displaystyle p^{\frac1{p-1}} \) in modulus.

  • (b) Prove that the polynomial \( P(x) \) is irreducible.


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