Example 3

Tag 06EG, permanent. Cite it as [Bourbaki, Tag 06EG].

By example 2 of IV, p. 64 and Prop. 1 above we thus have

\sum_{i=1}^n X_i^3 = s_{1,n}^3 - 3s_{1,n}s_{2,n} + 3s_{3,n}

for every integer

n \geq 3

. The commutativity of the diagram (10) moreover gives the formulae

X_1^3 + X_2^3 = s_{1,2}^3 - 3s_{1,2}s_{2,2},

X_1^3 = s_{1,1}^3.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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