8.3 Applications of the Fundamental Theorem

The Fundamental Theorem of Galois Theory links field extensions to group theory, revealing deep connections in algebra. This powerful tool helps us understand polynomial solvability, finite fields, and geometric constructions.

Applications of the theorem are far-reaching. We can determine which equations are solvable by radicals, analyze the structure of finite fields, and even prove the impossibility of certain geometric constructions using only a compass and straightedge.

Solvability of Polynomial Equations

Fundamental Theorem and Solvability

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• The Fundamental Theorem of Galois Theory establishes a correspondence between subfields of a splitting field and subgroups of the Galois group
• A polynomial equation is solvable by radicals if and only if its Galois group is solvable
  • Solvable means the group has a composition series with abelian factor groups
  • The Galois group of a polynomial is solvable if and only if the polynomial can be solved by radicals (using addition, subtraction, multiplication, division, and taking roots)

Insolvability of Higher Degree Equations

• The general polynomial equation of degree 5 or higher is not solvable by radicals
  • This is demonstrated by the insolvability of the quintic equation (fifth-degree polynomial)
  • Galois theory proves that the Galois group of a general quintic is not solvable, thus the equation cannot be solved by radicals
  • Examples of insolvable quintics include $x^5 - 4x + 2 = 0$ and $x^5 + 3x + 1 = 0$

Constructing Splitting Fields and Galois Groups

Splitting Fields

• A splitting field of a polynomial $f(x)$ over a field $F$ is the smallest field extension of $F$ that contains all the roots of $f(x)$
• To construct a splitting field, adjoin the roots of the polynomial to the base field successively until all roots are contained in the extension
  • Example: For $f(x) = x^3 - 2$ over $\mathbb{Q}$, adjoin $\sqrt[3]{2}$ to obtain the splitting field $\mathbb{Q}(\sqrt[3]{2})$

Galois Groups

• The Galois group of a polynomial $f(x)$ over a field $F$ is the group of automorphisms of the splitting field of $f(x)$ that fix $F$
• The Galois group can be determined by examining the permutations of the roots that preserve the relationships between the roots and the coefficients of the polynomial
  • Example: For $f(x) = x^4 - 2$ over $\mathbb{Q}$, the Galois group is the dihedral group $D_4$, generated by the permutations $(1234)$ and $(13)$
• The Fundamental Theorem of Galois Theory allows for the identification of intermediate fields and their corresponding subgroups of the Galois group

Structure of Finite Fields

Properties of Finite Fields

• Finite fields, also known as Galois fields, are fields with a finite number of elements
• The order of a finite field is always a prime power, denoted as $GF(p^n)$, where $p$ is a prime number and $n$ is a positive integer
• Every finite field is the splitting field of the polynomial $x^{p^n} - x$ over its prime subfield $GF(p)$

Galois Groups and Subfields of Finite Fields

• The Galois group of a finite field $GF(p^n)$ over its prime subfield $GF(p)$ is cyclic and generated by the Frobenius automorphism
• The subfields of a finite field correspond to the subgroups of its Galois group, which are also cyclic and of order dividing $n$
  • Example: $GF(2^6)$ has subfields $GF(2^3)$ and $GF(2^2)$ corresponding to subgroups of orders 2 and 3 in its Galois group
• The multiplicative group of a finite field is cyclic, and its generators are called primitive elements

Geometric Constructions and Field Extensions

Constructibility and Solvability

• Classical geometric construction problems involve constructing geometric figures using only a compass and straightedge
• The constructibility of a geometric figure is equivalent to the solvability of a corresponding polynomial equation by radicals
• A real number is constructible if and only if it lies in a field extension of the rationals obtained by a finite number of quadratic extensions
  • Example: $\sqrt{2}$ is constructible because it lies in the quadratic extension $\mathbb{Q}(\sqrt{2})$

Famous Construction Problems

• The Fundamental Theorem of Galois Theory can be used to determine the minimum field extension required for a given construction
• Famous construction problems, such as doubling the cube, trisecting an angle, and squaring the circle, can be proven impossible using Galois theory
  • Doubling the cube corresponds to solving the equation $x^3 = 2$, which is not solvable by radicals
  • Trisecting an angle corresponds to solving the equation $4x^3 - 3x - \cos(\theta) = 0$, which is not solvable by radicals for general angles $\theta$
  • Squaring the circle corresponds to constructing a square with the same area as a given circle, which is equivalent to constructing $\pi$, a transcendental number not contained in any finite extension of the rationals