Galois' Memoir: Lemma 4

The following is taken from the translation of Galois' Memoir by Harold M. Edwards found in his book Galois Theory. The proof itself is taken from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations.

Lemma 4:

Suppose one has formed the equation for V and that one has taken one of its irreducible factors so that V is the root of an irreducible equation.

Let V, V', V'', ... be the roots of this irreducible equation .

If a = f(V) is one of the roots of the given equation, f(V') will also be a root of the given equation.

In fact, we can show that:

r₁ = f₁(V), r₂ = f₂(V), ..., r_n = f_n(V)

Then it follows that for any root V' or V'' or ... the complete set of distinct roots is:

f₁(V'), f₂(V'), ..., f_n(V')

Proof:

(1) Let V be a Galois Resolvent of P(X) = 0. [see Lemma 2, here]

(2) Let r₁, ..., r_n be the roots of P(X).

(3) We know that for each r_i, there exists f_i [see Lemma 3, here] such that:

r_i = f_i(V)

with f_i(X) ∈ F(X)

(4) Let V₁ = V, V₂, ..., V_m be the roots of the minimum polynomial of V over F [see Theorem 1, here] which are in F(r₁, ..., r_n) [see Corollary 1.1, here]

(5) Since r_i = f_i(V), we have P(f_i(V)) = 0.

(6) If we view P(f_i(X)) as a polynomial, then, we have P(f_i(V_j)) = 0 for j = 1, ..., m since P(f_i(X)) and the minimal polynomial of V over F share at least one root V. [see Theorem, here]

(7) This shows that each f_i(V_j) must correspond to a root r_i since P(X)=0 if and only if X is a root.

(8) Now, I will end this proof by showing that for any root V_i, that the n roots are:

f₁(V_i), f₂(V_i), ..., f_n(V_i)

(9) Now we know that for V₁, f₁(V₁) ≠ f₂(V₁) ≠ ... ≠ f_n(V₁) [from step #3 above]

(10) Assume that f_u(V_i) = f_v(V_i)

where i ≠ 1

(11) Then V_j is a root of the polynomial f_u - f_v.

(12) But then all V₁, ..., V_m must likewise be roots of f_u - f_v. [see see Theorem, here]

(13) But this also means that V₁ is a root of f_u - f_v

(14) So it follows that f_u(V₁) = f_v(V₁)

(15) But from step #9 above this is only possible if u = v.

(16) Hence, we have shown that for any V_i,

f₁(V_i), ..., f_n(V_i) must be the n distinct roots [since they cannot be equal and each one is equal to a root.]

QED

References

• Harold M. Edwards, Galois Theory, Springer, 1984
• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001