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Is it possible to solve all fif the degree equations by radicals? Who or prove disproved this?

Is it possible to solve all fif the degree equations by radicals? Who or prove disproved this?

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**Question:** Is it possible to solve all fifth degree equations by radicals? Who proved or disproved this? **Answer:** The question touches on a significant area of algebraic theory concerning the solvability of polynomial equations by radicals. In mathematics, solving an equation "by radicals" means finding its roots using a finite sequence of operations involving addition, subtraction, multiplication, division, and the extraction of roots (square roots, cube roots, etc.). The answer to this question lies in the work of several mathematicians, most notably Évariste Galois and Niels Henrik Abel. **Abel's Theorem:** Niels Henrik Abel, a Norwegian mathematician, proved in the early 19th century that there is no general solution by radicals to polynomial equations of degree five or higher. This is known as the Abel-Ruffini theorem. The implication of Abel's work is that there is no formula that expresses the roots of all quintic (fifth degree) polynomials as a function of the coefficients using just arithmetic operations and radicals. **Galois Theory:** The work of Évariste Galois further developed these ideas by introducing what is now known as Galois theory. Galois showed how to determine whether a particular polynomial equation can be solved by radicals by studying the symmetries of its roots, encapsulated in what is called the "Galois group" of the polynomial. Together, Abel and Galois laid the foundation for a deep understanding of the solvability of polynomial equations, providing key insights into why some equations can be solved by radicals and others cannot.
Transcribed Image Text:**Question:** Is it possible to solve all fifth degree equations by radicals? Who proved or disproved this? **Answer:** The question touches on a significant area of algebraic theory concerning the solvability of polynomial equations by radicals. In mathematics, solving an equation "by radicals" means finding its roots using a finite sequence of operations involving addition, subtraction, multiplication, division, and the extraction of roots (square roots, cube roots, etc.). The answer to this question lies in the work of several mathematicians, most notably Évariste Galois and Niels Henrik Abel. **Abel's Theorem:** Niels Henrik Abel, a Norwegian mathematician, proved in the early 19th century that there is no general solution by radicals to polynomial equations of degree five or higher. This is known as the Abel-Ruffini theorem. The implication of Abel's work is that there is no formula that expresses the roots of all quintic (fifth degree) polynomials as a function of the coefficients using just arithmetic operations and radicals. **Galois Theory:** The work of Évariste Galois further developed these ideas by introducing what is now known as Galois theory. Galois showed how to determine whether a particular polynomial equation can be solved by radicals by studying the symmetries of its roots, encapsulated in what is called the "Galois group" of the polynomial. Together, Abel and Galois laid the foundation for a deep understanding of the solvability of polynomial equations, providing key insights into why some equations can be solved by radicals and others cannot.
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