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Definition 15.11.1. A henselian pair is a pair $(A, I)$ satisfying

  1. $I$ is contained in the Jacobson radical of $A$, and

  2. for any monic polynomial $f \in A[T]$ and factorization $\overline{f} = g_0h_0$ with $g_0, h_0 \in A/I[T]$ monic generating the unit ideal in $A/I[T]$, there exists a factorization $f = gh$ in $A[T]$ with $g, h$ monic and $g_0 = \overline{g}$ and $h_0 = \overline{h}$.

Comments (4)

Comment #457 by Kestutis Cesnavicius on

I would replace 'radical ideal' by 'Jacobson radical' for clarity.

Comment #459 by on

Well, I changed it in this section, because it is clearer as you say. But there are other locations where we use the language "the radical of " which I did not change. For example in Nakayama's lemma 10.20.1.

You can find the change here.

Comment #5687 by Laurent Moret-Bailly on

Condition (1) implies uniqueness of the decomposition in (2). It would be nice to mention it; in fact I would not be suprised if this were used somewhere later.

Comment #5762 by on

Going to leave as is for now. Some uniqueness is mentined in Lemma 15.11.6.

There are also:

  • 5 comment(s) on Section 15.11: Henselian pairs

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View Definition 15.11.1 as pdf