7.1 Galois groups of polynomials and field extensions

Galois groups are key to understanding polynomial roots and field extensions. They reveal symmetries in algebraic structures, showing how roots are related and which field elements stay fixed under certain transformations. This concept bridges abstract algebra and number theory.

The Fundamental Theorem of Galois Theory connects subgroups of Galois groups to intermediate fields. This link helps classify field extensions, determine solvability of equations, and uncover hidden relationships between different mathematical objects. It's a cornerstone of modern algebra.

Galois groups of polynomials and field extensions

Definition and properties of Galois groups

• The Galois group of a polynomial $f(x)$ over a field $F$, denoted $Gal(f/F)$, is the group of automorphisms of the splitting field of $f(x)$ that fix the base field $F$
• The Galois group of a field extension $L/K$, denoted $Gal(L/K)$, is the group of automorphisms of $L$ that fix the base field $K$
• The elements of the Galois group permute the roots of the polynomial while preserving the field operations and fixing the elements of the base field
  • For example, if $f(x) = x^3 - 2$ over $\mathbb{Q}$, the Galois group permutes the roots $\sqrt[3]{2}, \omega\sqrt[3]{2}, \omega^2\sqrt[3]{2}$, where $\omega$ is a primitive cube root of unity
• The Galois group is a subgroup of the symmetric group on the roots of the polynomial
  • In the previous example, the Galois group is isomorphic to $S_3$, the symmetric group on three elements
• The order of the Galois group divides the degree of the splitting field extension over the base field

Examples and applications of Galois groups

• The Galois group of a quadratic polynomial $ax^2 + bx + c$ over $\mathbb{Q}$ is:
  • Trivial if the discriminant $b^2 - 4ac$ is a perfect square in $\mathbb{Q}$
  • Isomorphic to $\mathbb{Z}/2\mathbb{Z}$ if the discriminant is not a perfect square
• The Galois group of a cyclotomic extension $\mathbb{Q}(\zeta_n)$ over $\mathbb{Q}$ is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{\times}$, the multiplicative group of integers modulo $n$
  • For example, the Galois group of $\mathbb{Q}(\zeta_5)$ over $\mathbb{Q}$ is isomorphic to $(\mathbb{Z}/5\mathbb{Z})^{\times} \cong \mathbb{Z}/4\mathbb{Z}$
• The Galois group of a finite field extension is always a finite group, and its order is equal to the degree of the extension
  • For instance, the Galois group of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$ is a cyclic group of order $n$

Determining Galois groups

Techniques for finding Galois groups

• To find the Galois group of a polynomial, first determine its splitting field by finding all the roots of the polynomial
• Identify the automorphisms of the splitting field that fix the base field, which form the Galois group
  • These automorphisms can be found by examining the action on the roots of the polynomial or on the generators of the field extension
• For simple extensions, the Galois group can be determined by examining the minimal polynomial of the extension
  • If the minimal polynomial has distinct roots, the Galois group is a subgroup of the symmetric group on the roots
• Determine the Galois group by examining the action of automorphisms on the roots of the polynomial or on the generators of the field extension

Examples of determining Galois groups

• The Galois group of $x^4 - 2$ over $\mathbb{Q}$ is the dihedral group of order 8, $D_4$
  • The splitting field is $\mathbb{Q}(\sqrt[4]{2}, i)$, and the automorphisms are determined by their action on $\sqrt[4]{2}$ and $i$
• The Galois group of $x^3 - 3x + 1$ over $\mathbb{Q}$ is the symmetric group $S_3$
  • The polynomial has three distinct real roots, and any permutation of these roots defines an automorphism of the splitting field
• The Galois group of $x^p - x - 1$ over $\mathbb{F}_p$ is a cyclic group of order $p$
  • The polynomial is irreducible over $\mathbb{F}_p$, and its splitting field is a degree $p$ extension of $\mathbb{F}_p$

Galois groups and splitting fields

Relationship between Galois groups and splitting fields

• The splitting field of a polynomial $f(x)$ over a field $F$ is the smallest field extension of $F$ containing all the roots of $f(x)$
• The Galois group of a polynomial acts transitively on the roots of the polynomial in the splitting field
  • This means that for any two roots $\alpha$ and $\beta$, there exists an automorphism in the Galois group that maps $\alpha$ to $\beta$
• The fixed field of the Galois group (elements unchanged by all automorphisms in the group) is precisely the base field

Galois correspondence

• The Galois correspondence establishes a one-to-one correspondence between the subgroups of the Galois group and the intermediate fields between the base field and the splitting field
  • The correspondence reverses inclusions: if $H \leq G$ are subgroups of the Galois group, then their fixed fields satisfy $Fix(G) \subseteq Fix(H)$
  • For example, if $L/K$ is a Galois extension with Galois group $G$, and $H$ is a subgroup of $G$, then the fixed field $L^H$ is an intermediate field between $K$ and $L$
• The Galois correspondence allows the study of field extensions by examining the subgroup structure of their Galois groups

Fundamental Theorem of Galois Theory

Statement and implications of the theorem

• The Fundamental Theorem of Galois Theory states that for a Galois extension $L/K$ with Galois group $G$, there is a one-to-one correspondence between the subgroups of $G$ and the intermediate fields between $K$ and $L$
  • The correspondence associates each subgroup $H$ of $G$ with its fixed field $L^H = {x \in L : \sigma(x) = x \text{ for all } \sigma \in H}$
  • The correspondence associates each intermediate field $M$ with its automorphism group $Aut(L/M) = {\sigma \in G : \sigma(x) = x \text{ for all } x \in M}$
• The Fundamental Theorem allows the classification of all intermediate fields of a Galois extension using the subgroup structure of the Galois group
• The degree of an intermediate field over the base field equals the index of its corresponding subgroup in the Galois group

Galois extensions and the Fundamental Theorem

• An extension $L/K$ is Galois if and only if it is normal (every irreducible polynomial in $K[x]$ that has a root in $L$ splits completely in $L$) and separable (every element in $L$ is separable over $K$)
  • For example, the splitting field of a separable polynomial is always a Galois extension
• The Galois group of a finite Galois extension is always isomorphic to a subgroup of the symmetric group on the roots of a defining polynomial for the extension
  • This follows from the fact that the Galois group permutes the roots of the defining polynomial
• The Fundamental Theorem provides a powerful tool for studying the structure of Galois extensions and their intermediate fields