9.2 Galois' criterion for solvability by radicals

Galois' criterion links polynomial solvability to group theory. It states a polynomial is solvable by radicals if and only if its Galois group is solvable. This powerful tool helps determine which equations have algebraic solutions.

Understanding Galois' criterion is key to grasping the limits of algebraic solutions. It explains why we can't solve all polynomials with radicals and connects abstract algebra to concrete mathematical problems we've struggled with for centuries.

Galois' Criterion for Solvability

Statement of Galois' Criterion

• Galois' criterion states that a polynomial $f(x) \in F[x]$ is solvable by radicals over a field $F$ if and only if its Galois group $Gal(f/F)$ is solvable
• A group $G$ is solvable if it has a subnormal series $G = G_0 \supset G_1 \supset ... \supset G_n = \{e\}$ such that each quotient group $G_i/G_{i+1}$ is abelian for $i = 0, 1, ..., n-1$

Key Concepts in Galois' Criterion

• The splitting field of a polynomial $f(x)$ over $F$ is the smallest field extension of $F$ that contains all the roots of $f(x)$
• The Galois group $Gal(f/F)$ is the group of automorphisms of the splitting field of $f(x)$ that fix the base field $F$
• Solvability by radicals means that the roots of the polynomial can be expressed using the four basic arithmetic operations ($+$, $-$, $\times$, $\div$) and taking $n$th roots (radicals) of elements in the base field $F$

Applying Galois' Criterion

Determining Solvability by Radicals

• To apply Galois' criterion, first determine the Galois group $Gal(f/F)$ of the polynomial $f(x)$ over the field $F$
• Examine the structure of the Galois group to determine if it is solvable, i.e., if it has a subnormal series with abelian quotient groups
  • Common solvable groups include abelian groups, cyclic groups, dihedral groups of order $2n$ for odd $n$, and the alternating group $A_4$
  • Common non-solvable groups include the symmetric groups $S_n$ for $n \geq 5$ and the alternating groups $A_n$ for $n \geq 5$
• If the Galois group is solvable, the polynomial is solvable by radicals. If the Galois group is not solvable, the polynomial is not solvable by radicals

Examples of Applying Galois' Criterion

• For example, the general polynomial of degree $n$ is solvable by radicals for $n \leq 4$ because its Galois group is a subgroup of $S_4$, which is solvable
• However, for $n \geq 5$, the general polynomial is not solvable by radicals because its Galois group contains $A_5$, which is not solvable
• The cyclotomic polynomial $\Phi_n(x)$ is always solvable by radicals because its Galois group is abelian (isomorphic to $(\mathbb{Z}/n\mathbb{Z})^\times$)
• The polynomial $x^5 - 4x + 2$ is not solvable by radicals over $\mathbb{Q}$ because its Galois group is isomorphic to $S_5$

Galois Group and Solvability

Relationship between Galois Group and Solvability

• The Galois group encodes the symmetries of the roots of a polynomial, and its structure determines whether the polynomial is solvable by radicals
• If the Galois group is solvable, the roots can be expressed in terms of radicals because the extension fields in the splitting field can be constructed by adjoining radicals successively
• If the Galois group is not solvable, there is no way to express the roots using radicals because the extension fields cannot be constructed by adjoining radicals

Consequences of Galois' Criterion

• The solvability of the Galois group is related to the existence of a radical extension of the base field that contains all the roots of the polynomial
• The unsolvability of the general quintic polynomial by radicals is a consequence of the fact that its Galois group, $S_5$, is not solvable
• The Abel-Ruffini theorem, which states that there is no general algebraic solution for polynomials of degree 5 or higher, follows from the unsolvability of $S_5$
• Galois' criterion provides a powerful tool for determining the solvability of polynomials and understanding the limitations of algebraic solutions

Necessity and Sufficiency of Galois' Criterion

Proving Necessity

• To prove necessity, assume the polynomial $f(x)$ is solvable by radicals over $F$
• Show that its splitting field can be obtained by successively adjoining radicals, and use this to construct a subnormal series of the Galois group with abelian quotient groups
• The key steps involve expressing the roots in terms of radicals and using the properties of radical extensions to build the subnormal series

Proving Sufficiency

• To prove sufficiency, assume the Galois group $Gal(f/F)$ is solvable, and use the subnormal series to construct a radical extension of $F$ that contains all the roots of $f(x)$
  • Use the Fundamental Theorem of Galois Theory to establish a correspondence between the intermediate fields of the splitting field and the subgroups of the Galois group
  • Show that the abelian quotient groups in the subnormal series correspond to radical extensions of the intermediate fields
• The proof relies on the properties of solvable groups, the Fundamental Theorem of Galois Theory, and the correspondence between field extensions and subgroups of the Galois group

Significance of Necessity and Sufficiency

• The necessity and sufficiency of Galois' criterion demonstrate the deep connection between the algebraic structure of the Galois group and the solvability of polynomials by radicals
• The equivalence between solvability by radicals and having a solvable Galois group highlights the fundamental role of Galois theory in understanding the nature of algebraic equations
• The proof of Galois' criterion showcases the power and elegance of the Fundamental Theorem of Galois Theory and its applications in solving classical problems in algebra