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Chapter 12: Problem 1

In Exercises 1 through 8 express the splitting field of the indicated polynomial \(f(x) \in \mathbb{Q}[x]\) as a radical extension of \(\mathrm{Q}\). $$ x^{2}+2 x+2 $$

Short Answer

Expert verified
The splitting field is \(\mathbb{Q}(i)\).

Step by step solution

01

Determine the Roots

The polynomial is \( x^2 + 2x + 2 \). To find the roots, use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = 2 \). Calculate the discriminant: \( b^2 - 4ac = 4 - 8 = -4 \).
02

Simplify the Roots

Because the discriminant is \(-4\), the expression under the square root is negative, indicating complex roots. Compute: \[ x = \frac{-2 \pm \sqrt{-4}}{2} = \frac{-2 \pm 2i}{2} = -1 \pm i \].
03

Express the Splitting Field

The roots are \(-1 + i\) and \(-1 - i\), both of which require \(i\), the imaginary unit. The minimal field extension needed to express the roots is \(\mathbb{Q}(i)\), because \(i\) cannot be represented with just rational numbers.
04

Verify the Splitting Field

Check that \(\mathbb{Q}(i)\) is indeed a splitting field. The polynomial completely factors over \(\mathbb{Q}(i)\) as \((x - (-1 + i))(x - (-1 - i)) = 0\). Therefore, \(\mathbb{Q}(i)\) is a splitting field over \(\mathbb{Q}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Roots
At the heart of many algebraic problems lies finding the roots of a polynomial. A polynomial root is a solution to the equation formed by setting the polynomial to zero. For a simple quadratic polynomial like \(x^2 + 2x + 2\), finding the roots involves finding those values of \(x\) for which the equation equals zero.
  • Roots can be real or complex, depending on the discriminant (\(b^2 - 4ac\)). A positive discriminant indicates real roots, zero indicates a repeated root, and negative indicates complex roots.
  • In this context, the polynomial has complex roots due to a negative discriminant.
  • These roots often reveal fundamental properties of the polynomial and the fields from which it originates.
Understanding polynomial roots is crucial because they tell us about the behavior of the polynomial over different number systems, such as real or complex numbers.
Quadratic Formula
The quadratic formula is a handy tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). By using it, we can find the precise roots of any quadratic polynomial.
  • Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
  • This formula requires calculating the discriminant \(b^2 - 4ac\), which informs us whether the roots are real or complex.
  • In our case, with \(a = 1\), \(b = 2\), and \(c = 2\), we found the discriminant to be negative, resulting in complex roots.
This formula not only aids in finding roots but also in understanding the types of roots we are dealing with, based on the nature of the discriminant.
Complex Numbers
When dealing with negative discriminants in quadratic equations, we encounter complex numbers.
  • Complex numbers extend the idea of "number" to include all sums of real numbers and imaginary numbers (i.e., numbers formed with the unit \(i\) such that \(i^2 = -1\)).
  • The roots found in this problem, \(-1 \pm i\), are examples of complex numbers. Here, \(-1\) is the real part, and \(\pm i\) is the imaginary part.
  • Complex numbers are arranged on a two-dimensional plane called the complex plane, where real numbers map onto the x-axis and imaginary numbers onto the y-axis.
Recognizing and working with complex numbers is crucial in higher mathematics because they arise naturally in many areas of study.
Field Extension
In order to express the roots of polynomials fully, especially when they are not real, we utilize field extensions.
  • A field extension expands a base field to include additional elements necessary to express roots or other mathematical expressions. In this problem, the base field is \(\mathbb{Q}\), the field of rational numbers.
  • To accommodate the complex roots \(-1 \pm i\), we extend \(\mathbb{Q}\) to \(\mathbb{Q}(i)\), a field containing all numbers of the form \(a + bi\) where \(a, b\) are rational numbers.
  • This minimal extension necessary to express the polynomial's roots completely is called the splitting field.
Field extensions are an essential concept in algebra, allowing us to explore solutions beyond the constraints of simpler number systems such as \(\mathbb{Q}\).

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Most popular questions from this chapter

For Exercises 23 through 26 consider \(E=\mathbb{Q}\left(2^{1 / 3}, \sqrt{3} i\right)\) over \(Q\) and calculate the norm and trace of the indicated elements \(\alpha \in E\). $$ \alpha=2^{1 / 3} $$

In Exercises 1 through 7 determine for the indicated \(n\) whether or not the regular \(n\) -gon is constructible by straightedge and compass. $$ n=52 $$

In Exercises 12 through 17 determine whether or not the indicated quintic polynomial in \(Q[x]\) is solvable by radicals over \(Q\). $$ x^{5}+1 $$

Calculate the \(\mathrm{Gal}(E / \mathrm{Q})\). where \(E\) is the splitting field in \(\mathrm{C}\) of the indicated polynomial \(f(x) \in \mathbb{Q}[x]\). $$ f(x)=x^{4}-2 x $$

Calculate the Galois group \(\mathrm{Gal}(E / Q)\) for the indicated fields \(E\) \(E=\mathbb{Q}(\omega),\) where \(\omega\) is the primitive cube root of unity
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