Question

(i) Prove Eisenstein's theorem: If \(f \in Z[x]\) and \(p \in N\) is a prime number such that \(p \nmid l c(f)\), \(p\) divides all other coefficients of \(f\), and \(p^{2} \uparrow f(0)\), then \(f\) is irreducible in \(\mathbb{Q}[x]\). (ii) Conclude that for any \(n \in N\), the polynomial \(x^{n}-p\) is irreducible in \(Q[x]\).

Short answer

By Eisenstein's criterion, \(x^n - p\) is irreducible in \(\mathbb{Q}[x]\).

Step-by-step solution

Understanding the Problem

We want to prove Eisenstein's criterion for determining irreducibility of a polynomial over \(\mathbb{Q}[x]\). The theorem gives conditions involving a prime number \(p\) and coefficients of a polynomial \(f\) to establish irreducibility in \(\mathbb{Q}[x]\). Once proven, utilize these conditions to show that \(x^n - p\) is irreducible for any natural number \(n\) when \(p\) is prime.

Restate Eisenstein's Theorem Hypotheses

Consider a polynomial \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0\) in \(\mathbb{Z}[x]\). Assume there exists a prime \(p\) such that \(p mid a_n\), \(p \mid a_i\) for \(i = 0, 1, \ldots, n-1\), and \(p^2 mid a_0\).

Prove Irreducibility Using Eisenstein's Criterion

Assume for contradiction that \(f(x)\) is reducible over \(\mathbb{Q}[x]\). Then it can be factorized as \(f(x) = g(x)h(x)\) with non-constant polynomials \(g, h \in \mathbb{Q}[x]\). Clearing denominators, we write \(g(x) = b_kx^k + \cdots + b_0\) with gcd of coefficients 1. Similarly for \(h(x)\). Multiply to match terms of \(f(x)\): all leading coefficients must multiples of \(p\), which contradicts \(p mid a_n\). For constant term \(b_0\), we derive a contradiction owing to \(p^2 mid a_0\).

Apply Eisenstein's Criterion to \(x^n - p\)

Consider \(f(x) = x^n - p\). Here, \(a_n = 1\) (not divisible by any \(p\)), \(a_i = 0\) thus trivially divisible by \(p\) for \(n-1 \geq i \geq 1\), \(a_0 = -p\) (divisible by \(p\) but not by \(p^2\)). Therefore, by Eisenstein's criterion, \(f(x) = x^n - p\) is irreducible in \(\mathbb{Q}[x]\).

Key concepts

Irreducible Polynomials

An irreducible polynomial is a polynomial that cannot be factored into the product of two non-constant polynomials with coefficients in a given field or ring, such as the integers \( \mathbb{Z} \) or rationals \( \mathbb{Q} \). When a polynomial is irreducible in a given ring, it shares similar properties to a prime number, which cannot be divided evenly by any number other than itself and one. Determining irreducibility is crucial in many areas of mathematics, such as number theory and algebra.

Eisenstein's Criterion is a major tool for proving irreducibility. It sets specific conditions involving a prime number that, when satisfied, ensure that a polynomial is irreducible over \( \mathbb{Q}[x] \). By using this criterion, you can prove that a polynomial doesn't break down into smaller components, much like verifying a number is prime by showing it has no divisors other than itself and one. Understanding irreducibility helps in simplifying expressions and in finding roots of equations, which is essential for solving algebraic problems efficiently.

Prime Numbers in Algebra

Prime numbers are the building blocks of integers, and they play a crucial role in algebra, especially in the factorization of polynomials. A prime number is an integer greater than one that cannot be formed by multiplying two smaller natural numbers. Just as prime numbers are indivisible by other numbers, in algebra, primes are used to simplify and verify properties of polynomials.

Eisenstein's Criterion utilizes primes to set conditions for the irreducibility of polynomials. Specifically, it requires the use of a prime number \( p \) to divide all coefficients of a polynomial (except the leading one) and a stricter condition at the constant term. Through this process, the prime helps to verify that the polynomial cannot be decomposed into simpler factors over the rationals \( \mathbb{Q}[x] \).

• Prime \( p \) ensures no division into smaller parts over rationals.
• Key tool for proving polynomial characteristics.

Prime numbers in algebra help establish foundational properties of algebraic structures and enable deeper exploration into the nature of functions and equations.

Polynomial Factorization

Factorization involves decomposing a polynomial into a product of simpler polynomials that, when multiplied together, give the original polynomial. This process is akin to breaking down numbers into their prime factors. Just as understanding the prime factorization of a number can reveal much about its properties, polynomial factorization can offer insight into the structure and characteristics of a polynomial.

For instance, Eisenstein's Criterion aids in identifying when a polynomial remains indivisible (irreducible) over \( \mathbb{Q}[x] \). By applying the criterion, if a polynomial meets specific conditions set by a prime, it remains in its simplest form within the rationals, indicating it cannot be factored further.

• Key to simplifying problems in algebra.
• Helps in plotting function behavior and finding roots.

Therefore, factorization is not merely an algebraic manipulation but a critical process for understanding and solving complex problems in mathematics, making it an essential concept for students to grasp.