7.3 Computing Galois groups

Computing Galois groups is a crucial skill in Galois theory. It involves finding the automorphisms of a splitting field that fix the base field. This process helps us understand the structure of field extensions and their symmetries.

Galois groups connect polynomials to field theory and group theory. By computing them, we can determine if equations are solvable by radicals and uncover deep relationships between different mathematical structures. It's a powerful tool for solving algebraic problems.

Galois Groups of Polynomials

Splitting Fields and Galois Groups

• The splitting field of a polynomial $f(x)$ over a field $F$ is the smallest field extension of $F$ that contains all the roots of $f(x)$
• The Galois group of $f(x)$ over $F$, denoted $Gal(f/F)$ or $Gal(E/F)$ where $E$ is the splitting field, is the group of automorphisms of the splitting field that fix the base field $F$
  • Example: For the polynomial $f(x) = x^3 - 2$ over $\mathbb{Q}$, the splitting field is $\mathbb{Q}(\sqrt[3]{2}, \omega\sqrt[3]{2}, \omega^2\sqrt[3]{2})$, where $\omega$ is a primitive third root of unity
• The Galois group is a subgroup of the symmetric group $S_n$, where $n$ is the degree of $f(x)$

Computing Galois Groups using Splitting Fields

• To compute the Galois group, first find the splitting field by adjoining the roots of $f(x)$ to the base field $F$
  • Example: For $f(x) = x^4 - 2$ over $\mathbb{Q}$, adjoin $\sqrt[4]{2}$, $i\sqrt[4]{2}$, $-\sqrt[4]{2}$, and $-i\sqrt[4]{2}$ to $\mathbb{Q}$ to obtain the splitting field
• Determine the automorphisms of the splitting field that fix $F$
  • These automorphisms can be found by examining the permutations of the roots that preserve the coefficients of $f(x)$
  • Example: For $f(x) = x^4 - 2$, the automorphisms are generated by $\sqrt[4]{2} \mapsto i\sqrt[4]{2}$ and $\sqrt[4]{2} \mapsto -\sqrt[4]{2}$, forming the dihedral group $D_4$

Galois Groups of Field Extensions

Minimal Polynomials and Galois Groups

• The minimal polynomial of an element $\alpha$ over a field $F$ is the monic polynomial of lowest degree with coefficients in $F$ that has $\alpha$ as a root
• If $E = F(\alpha)$ is a field extension, the minimal polynomial of $\alpha$ over $F$ can be used to determine the Galois group $Gal(E/F)$
  • The degree of the minimal polynomial equals the degree of the field extension $[E:F]$
  • The roots of the minimal polynomial are the conjugates of $\alpha$ under the action of the Galois group
• By examining the permutations of the roots that preserve the coefficients of the minimal polynomial, the Galois group can be determined
  • Example: For $\alpha = \sqrt{2} + \sqrt{3}$ over $\mathbb{Q}$, the minimal polynomial is $x^4 - 10x^2 + 1$, and the Galois group is the Klein four-group $V_4$

Galois Extensions and Fixed Fields

• An extension $E/F$ is Galois if and only if $F$ is the fixed field of $Gal(E/F)$
  • The fixed field of a subgroup $H$ of $Gal(E/F)$ is the set of elements in $E$ that are fixed by every automorphism in $H$
• Properties of field extensions (such as normality and separability) and their corresponding subgroups can be used to compute Galois groups
  • Example: If $E/F$ is a Galois extension and $[E:F] = p$ is prime, then $Gal(E/F)$ is cyclic of order $p$

Galois Group Computation Techniques

Fundamental Theorem of Galois Theory

• The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between the intermediate fields of a Galois extension and the subgroups of its Galois group
  • For a Galois extension $E/F$, there is a bijection between the intermediate fields $K$ (with $F \subseteq K \subseteq E$) and the subgroups $H$ of $Gal(E/F)$
• This correspondence allows for the computation of Galois groups by examining the lattice of intermediate fields
  • Example: For the splitting field of $x^4 - 2$ over $\mathbb{Q}$, the lattice of intermediate fields corresponds to the subgroup lattice of $D_4$

Galois Correspondence

• The Galois correspondence relates the lattice of intermediate fields to the lattice of subgroups of the Galois group
  • If $K$ is an intermediate field, then $Gal(E/K)$ is a subgroup of $Gal(E/F)$
  • If $H$ is a subgroup of $Gal(E/F)$, then the fixed field of $H$ is an intermediate field
• This correspondence can be used to determine the Galois group by analyzing the intermediate fields and their corresponding subgroups
  • Example: For the splitting field of $x^3 - 2$ over $\mathbb{Q}$, the intermediate fields $\mathbb{Q}(\sqrt[3]{2})$ and $\mathbb{Q}(\omega)$ correspond to subgroups of order 2 and 3, respectively, in the Galois group $S_3$

Degree vs Order in Galois Theory

Degree of Splitting Field and Order of Galois Group

• The degree of the splitting field $E$ over the base field $F$, denoted $[E:F]$, is equal to the order (number of elements) of the Galois group $Gal(E/F)$
  • This relationship is a consequence of the Fundamental Theorem of Galois Theory
• If $f(x)$ is a separable polynomial of degree $n$ over $F$, then the splitting field $E$ has degree dividing $n!$ over $F$
  • Example: For $f(x) = x^3 - 2$ over $\mathbb{Q}$, the splitting field has degree 6, which divides $3! = 6$

Galois Groups as Subgroups of Symmetric Groups

• The Galois group of a separable polynomial $f(x)$ is a subgroup of the symmetric group $S_n$, and its order divides $n!$
• Understanding this relationship can help in determining the possible Galois groups for a given polynomial or field extension
  • Example: For a separable quartic polynomial, the Galois group must be a subgroup of $S_4$, which has order dividing $4! = 24$, narrowing down the possible Galois groups to $S_4$, $A_4$, $D_4$, $V_4$, or $C_4$