Lagrange and the Theory of Equations: Solving the Unsolvable, Study notes of Mathematics

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Beginnings of abstract algebra

Week 5 Lecture 10

New developments: a mathematics whose subject matter is no longer space or number

• theory of permutations
• recognition of abstract structures (groups, rings, fields,... )
• linear algebra

Recall that cubics can be solved by means of quadratics.

To solve x^3 + px = q

put u − v = q and uv = (p 3 )^3 to give

27 u^2 = p^3 + 27u

(then x = u

(^13)

• v

(^13) ).

Recall that quartics can be solved by means of cubics.

To solve x^4 = px^2 + qx + r we need to solve

8 y^3 + 4py62 + 8ry + (4pr − q^2 ) = 0 (Cardano)

or

y^6 − 2 py^4 + (p^2 − 4 r)y^2 − q^2 = 0 (Descartes).

Euler’s hypothesis (1733)

For an equation of degree n the degree of the reduced equation will be n − 1.

Bezout’s hypothesis (1764)

For an equation of degree n the degree of the reduced equation will in general be (n − 1)! (possibly further reducible to (n − 2)!)

Lagrange ‘Refl´exions sur la r´esolution alg´ebrique des ´equations’ (1771,1772)

Examined all known methods of solving

• cubics (Cardano, Tschirnhaus, Euler, Bezout)
• quartics (Cardano, Descartes, Tschirnhaus, Euler, Bezout)

Cubics:

roots of the resolvent are (multiples of) values of

y = x 1 + α^2 x 2 + αx 3

(y^3 takes 2 values as x 1 , x 2 , x 3 are permuted)

Quartics:

roots of the resolvent are (multiples of) values of

y = x 1 x 2 + x 3 x 4

or

z = (x 1 + x 2 ) − (x 3 + x 4 )

(y takes 3 values, z takes 6 values, as x 1 , x 2 , x 3 , x 4 are permuted)

Lagrange’s theorem:

Suppose an equation has n roots x 1 , x 2 , x 3 ,

...

In general, if f (x 1 , x 2 , x 3 ,.. .) is a solution of the resolvent, then the resolvent will be of degree n!

But suppose f is invariant under x 1 → x 2 , x 2 → x 3 , x 3 → x 1.

Then f (x 1 , x 2 , x 3 ,.. .), f (x 2 , x 3 , x 1 ,.. .), f (x 3 , x 1 , x 2 ,.. .) are equal roots, and the degree of the resolvent reduces to n 3!

Lagrange’s work on the number of values that a function could take under permutation of its variables deeply influenced

• Ruffini
• Cauchy

Paolo Ruffini Teoria general delle equazione (1799)

• showed that a function of 5 variables cannot take 3 or 4 values
• sent to Lagrange in 1802; no response
• further explanatory papers 1802, 1806

Cauchy M´emoire sur le nombre de valeurs qu’un fonction peut accq´erir... (written 1812; published 1815)

• let N be the number of a values of a function of n variables; if p ≤ n (p prime) either N = 2 or N ≥ p
• conjecture: for n ≥ 5, either N = 2 or N ≥ n (proved for n = 6)
• established theory and notation for ‘substitutions’

Groups