Set embedding of extension in QQ[2^(1/3)]

[vdelecroix]

Before

sage: QQ[2^(1/3)].coerce_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Real Lazy Field
  Defn: a -> 1.259921049894873?

After

sage: QQ[2^(1/3)].coerce_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Algebraic Real Field
  Defn: a -> 1.259921049894873?

[vdelecroix]

New commits:

​3360768

22150 set embedding appropriately in QQ[2^(1/3)]

[chapoton]

Needs rebase.

Moreover, there is a problem coming from

sage: sqrt(-3).is_real()
True

[mmezzarobba]

Whoops, I had forgotten about this ticket, but see #23117.

[chapoton]

so we can close this as duplicate, right ?

[chapoton]

Hmm. currently we get:

Generic morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Real Lazy Field
  Defn: a -> 1.259921049894873?

with target the real lazy field instead of AA.

[vdelecroix]

It would have been better if you just have read the ticket description.

[chapoton]

What do you mean ? The ticket description is no longer valid...

[vdelecroix]

The branch is not either. I think that #23117 was actually a bad idea. Coercion should not go through the real lazy field.

[mmezzarobba]

Replying to vdelecroix:

The branch is not either. I think that #23117 was actually a bad idea. Coercion should not go through the real lazy field.

I agree that complex embeddings should go to QQbar or something like that by default, but for me this is a separate issue, already tracked elsewhere.

However, having the K[α] syntax create the “standard” embedding makes more sense to me (less confusing, probably breaks fewer existing things) than having it attempt to embed the number field in α's parent or something like that—if that's what you have in mind.