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Galois theory has always struck me as rather mysterious, perhaps because its modern formulation is shrouded in concepts that did not yet exist during Galois' time (e.g., fields, groups, vector spaces). Even after taking a course on Galois theory, I have little idea of how Galois might have arrived upon his ideas in the first place.

Question: What was Galois' original line of reasoning that led him to the unsolvability of the quintic?

There is a great answer here that explains some of the differences between Galois' original work and modern Galois theory. However, it doesn't really explain how Galois' proof worked, or how he discovered it in the first place.

From what I gathered, Galois was interested in symmetric functions, which are rational functions in some indeterminates $t_1, \dots, t_n$ that are invariant under permutations of these indeterminates. The most important examples of symmetric functions are the coefficients of the polynomial $(x - t_1) \cdots (x - t_n)$.

Here is my guess as to how Galois proceeded. He considered the permutations of $t_1, \dots, t_n$ as a 'group' acting on the set of all rational functions. The symmetric functions are precisely those rational functions that are fixed by every permutation. Naturally, he began to consider the rational functions that are fixed by 'subgroups' of the permutation group (e.g., he may have considered the rational functions fixed by the subgroup of even permutations). From this, he eventually realized that there is a sort of correspondence between these subgroups and the operation of 'adjoining' certain formulas (e.g., he may have discovered that the subgroup of even permutations corresponds to adjoining the square root of the discriminant). This is essentially the fundamental theorem of Galois theory. Furthermore, he noticed that the well-known quadratic/cubic/quartic formulas are each given by successively adjoining roots of symmetric functions, which corresponds to a chain of subgroups of the permutation group. This led him to study the subgroups of the permutation group on five indeterminates, and he eventually realized that there is no chain of subgroups corresponding to a quintic formula.

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    $\begingroup$ The book Galois Theory by H.M. Edwards puts Galois' ideas inan historical context. It also contains Galois' original memoir. $\endgroup$
    – P. Lawrence
    Commented Jul 31, 2023 at 7:35
  • $\begingroup$ I also always had a hard time to understand this complicated stuff. The basic idea of Galois was to reduce questions about intermediate fields to questions about subgroups. The result that polynomials with degree $5$ or moe cannot generally be solved by radicals is the most beautiful , a long time it was tried to find such formulas, now we can save this time because we know it is pointless to search them. $\endgroup$
    – Peter
    Commented Jul 31, 2023 at 8:35
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    $\begingroup$ I'm not sure, but history of maths and science stackexchange may be a more suitable place to ask this question. $\endgroup$
    – Adam Rubinson
    Commented Jul 31, 2023 at 9:31
  • $\begingroup$ The key idea which Galois developed was that of Galois resolvent which is a rational function $V=f(r_1,r_2,\dots,r_n)$ of roots $r_i$ of a given polynomial such that each root $r_i$ can be expressed as a rational function of $V$ ie $r_i=\phi_i(V) $. This is essentially constructing the splitting field of the polynomial by adjoining $V$ to the field of coefficients of the polynomial. Galois then develops the notion of group of permutation of roots $r_i=\phi_i(V) $ and says that any function of roots which remains invariant under these permutations will be rationally known and conversely too. $\endgroup$
    – Paramanand Singh ♦
    Commented Aug 3, 2023 at 5:21
  • $\begingroup$ The Galois group of permutations of roots is explicitly determined by finding the minimal polynomial for $V$ of degree, say $m$, and if $V=V_1,V_2,\dots,V_m$ are roots of this minimal polynomial then the Galois groups consists of permutations $(\phi_1(V_1),\phi_2(V_1),\dots,\phi_n(V_1))$,$(\phi_1(V_2),\phi_2(V_2),\dots,\phi_n(V_2))$,$\dots$,$(\phi_1(V_m),\phi_2(V_m),\dots,\phi_n(V_m))$. The Galois group has order $m$ which is degree of minimal polynomial for $V$ and is thus the degree of splitting field over field of coefficients. $\endgroup$
    – Paramanand Singh ♦
    Commented Aug 3, 2023 at 5:29

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