Student Scholarship
Document Type
Research Paper
Abstract
This paper investigates the algebraic conjecture that the general equation of degree n is not solvable by radicals if n is greater than four. While the author does not intend to provide a new rigorous proof, the work serves to organize and present the fundamental mathematical concepts necessary to understand the proof for an undergraduate audience. The historical context of the problem is explored, noting the early achievements in solving cubic and quartic equations during the sixteenth century and the later contributions of mathematicians like Lagrange, who attempted to find a general reduction method for the quintic. The ultimate solution is attributed to Evariste Galois, who established that an algebraic equation is solvable by radicals if and only if its associated group is solvable.
To illustrate these concepts, the text defines and exemplifies the properties of polynomials, including the relationship between roots and coefficients. The discussion then transitions into group theory, covering essential structures such as symmetric groups, normal subgroups, and factor groups. These algebraic structures are linked to field theory, where the paper defines field extensions, splitting fields, and the degree of extensions.
The final sections of the paper apply these definitions to several important theorems, such as Kronecker’s Theorem and the Fundamental Theorem of Galois Theory. The author demonstrates the solvability of the symmetric groups for degrees two, three, and four, while providing a proof outline showing that the symmetric group on n letters is not solvable for n greater than four. Consequently, because the group of the general equation of degree n is the symmetric group, the paper concludes that such equations cannot be solved by radicals when the degree exceeds four.
Research Highlights
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The Problem: This paper investigates the conjecture that a general algebraic equation of degree $n$ is not solvable by radicals if $n > 4$.
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The Method: The author examines and exemplifies concepts from Galois theory, specifically focusing on the properties of polynomials, group theory, mappings (homomorphisms, isomorphisms, automorphisms), and field extensions.
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Qualitative Finding: An algebraic equation is solvable by radicals if and only if its associated group is solvable; the symmetric group $S_{n}$ is solvable for $n \leq 4$ but is not solvable for $n > 4$.
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Finding: The general equations of degree two, three, and four are solvable because their groups are solvable, whereas the general quintic and higher-degree equations are not solvable by radicals.
Publication Date
4-1973
Recommended Citation
Davis, Stephen, "The Solution of Equations by Radicals" (1973). Student Scholarship. 162.
https://digitalcommons.lindenwood.edu/student-research-papers/162
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
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