Errata

Example 5.8.8, page 136 and Exemple 5.8.8, page 128.
Of course, the least common multiple of 2, 3, and 5, is equal to 30, and not
to 60 as incorrectly written. One thus needs an additional argument to conclude that the Galois group of the indicated polynomial is equal to $\mathfrak A_5$.
However, conjugating a double transposition by a 5cycle, gives a different double transposition, so that this group contains at least two elements of order 2. Its cardinality is thus divisible by 4.
 The first sentence of statement of corollary 6.5.13 should read: “Let $A$ be a ring and let $M$ be a noetherian $A$module.”
In the proof, when I invoke corollary 6.5.13, one should of course
read “proposition 6.5.12”. [12 february 2023]

The given proof of theorem 9.5.2 (Seidenberg's Goingdown theorem).
makes use of proposition 9.5.1 which requires the extension $E\to F$
to be finite. Therefore, one should add to the hypothesis
of theorem 9.5.2 that the fraction field of $B$ is a
finite extension of the fraction field of $A$.
(This would hold, in particular, if $B$ is finitely generated over $A$
but that hypothesis is more difficult to check.)
See Bourbaki, Algèbre commutative, V, p. 46, prop. 6
for the general case.
[E. Kowalski, 17 november 2021]
Voir l'erratum cijoint (12 septembre 2022).
Antoine ChambertLoir