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Magazine Chapter 12: Problem 12
In Exercises 12 through 16 determine the Galois group over \(Q\) of the indicated cubic polynomial. $$ x^{3}-3 x+1 $$
Short Answer
Expert verified
The Galois group is \( S_3 \).
Step by step solution
01
Factoring the Polynomial
To find the Galois group, we first attempt to factor the polynomial. The polynomial given is \( x^3 - 3x + 1 \). Checking for rational roots using the Rational Root Theorem, we test the factors of 1: \( \pm 1 \). Neither of these roots satisfy the equation, so the polynomial is irreducible over \( \mathbb{Q} \).
02
Finding the Discriminant
Calculate the discriminant of the cubic polynomial \( x^3 - 3x + 1 \). The formula for the discriminant of \( ax^3 + bx^2 + cx + d \) is \( b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd \). Here, \( a = 1, b = 0, c = -3, d = 1 \). Substituting these values in, we find:\[ \Delta = 0^2(-3)^2 - 4 \cdot 1 \cdot (-3)^3 - 4 \cdot 0^3 \cdot 1 - 27 \cdot 1^2 \cdot 1^2 + 18 \cdot 0 \cdot (-3) \cdot 1 = 108 - 27 = 81 \].
03
Analyzing the Discriminant
The discriminant \( \Delta = 81 \), which is a positive perfect square (\( 9^2 \)). This indicates that the cubic polynomial has three distinct real roots.
04
Determining the Galois Group
Since the polynomial is irreducible over \( \mathbb{Q} \) and the discriminant is a positive perfect square, the Galois group of the polynomial is isomorphic to the symmetric group \( S_3 \). This means it is the full symmetric group on three elements, which is characteristic of a splitting field with three distinct roots.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cubic Polynomial
A cubic polynomial is a polynomial of degree three. In general, it can be written as \( ax^3 + bx^2 + cx + d \), where \( a eq 0 \). In the given problem, we've been given the polynomial \( x^3 - 3x + 1 \).
This is a simple form of a cubic polynomial because it lacks the \( x^2 \) term, which makes it easier to handle while applying various polynomial theorems.
This is a simple form of a cubic polynomial because it lacks the \( x^2 \) term, which makes it easier to handle while applying various polynomial theorems.
- The degree of the polynomial tells us how many roots (both real and complex) the polynomial can have.
- Because it is a cubic polynomial, it can have up to three real roots or a combination of real and complex roots.
Discriminant
The discriminant of a polynomial is a crucial value that provides insight into the nature of the roots of the polynomial. For a cubic polynomial \( ax^3 + bx^2 + cx + d \), the formula is given by:\[ b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd \].
In this example, the discriminant helps us determine if the roots are real or complex. In the case of \( x^3 - 3x + 1 \), where \( a = 1, b = 0, c = -3, d = 1 \),
In this example, the discriminant helps us determine if the roots are real or complex. In the case of \( x^3 - 3x + 1 \), where \( a = 1, b = 0, c = -3, d = 1 \),
- The discriminant comes out to be 81, which is a perfect square \((9^2)\).
- This indicates the polynomial has three distinct real roots.
Rational Root Theorem
The Rational Root Theorem is used to identify possible rational solutions of a polynomial equation. It states that any rational solution of the polynomial equation \( ax^n + bx^{n-1} + ... + d = 0 \) must be of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term \( d \) and \( q \) is a factor of the leading coefficient \( a \).
For \( x^3 - 3x + 1 \), we identify potential rational roots as \( \pm 1 \), derived from
For \( x^3 - 3x + 1 \), we identify potential rational roots as \( \pm 1 \), derived from
- The possible factors of \( 1 \) (from the constant term \( d \)).
- The possible factors of \( 1 \) (from the leading coefficient \( a \)).
Irreducible Polynomial
An irreducible polynomial is a polynomial that cannot be factored into a product of two non-constant polynomials over a given field. For a polynomial to be irreducible over \( \mathbb{Q} \), it should not factorize into simpler polynomials with rational coefficients.
In our given problem with the polynomial \( x^3 - 3x + 1 \),
In our given problem with the polynomial \( x^3 - 3x + 1 \),
- The Rational Root Theorem did not yield any rational roots.
- This absence of rational solutions implies the polynomial cannot be simplified or factored further within \( \mathbb{Q} \).
