Cyclotomic embeddings should respect coercions.

[robertwb]

sage: zC=(CC(-1)^(1/3))^2
sage: Ka.<aa>=CyclotomicField(3,embedding=zC)
sage: Kb.<bb>=CyclotomicField(3,embedding=zC^2)
sage: Kc.<cc>=CyclotomicField(3,embedding=CC(17))
sage: CC(aa)
-0.500000000000000 + 0.866025403784439*I
sage: CC(bb)
-0.500000000000000 - 0.866025403784439*I
sage: CC(Ka(bb))
-0.500000000000000 + 0.866025403784439*I

It's better to disallow coercions between these then allow this nonsense

sage: CC(aa+bb)
 -1.00000000000000 + 1.73205080756888*I

sage: CC(bb+aa)
 -1.00000000000000 - 1.73205080756888*I

apply trac-13765-cyclo-embeddings.patch

CC: @nbruin

Component: number fields

Author: Robert Bradshaw

Reviewer: Frédéric Chapoton, Stephan Ehlen

Merged: sage-5.7.beta2

Issue created by migration from https://trac.sagemath.org/ticket/13765

[robertwb]

Author: Robert Bradshaw

[fchapoton]

comment:3

One minor comment about the doc :

maybe you could use the trac role :trac:13765 to link to the ticket ?

(see #12490)

[fchapoton]

comment:4

There is a typo "no such integer exits"

[fchapoton]

comment:5

I have made the small changes that I suggested :

apply trac-13765-cyclo-embeddings.patch

[robertwb]

comment:6

Your changes look fine to me, thanks.

[fchapoton]

comment:7

apply trac-13765-cyclo-embeddings.patch

[fchapoton]

Description changed:

--- 
+++ 
@@ -22,3 +22,5 @@
  -1.00000000000000 - 1.73205080756888*I
 ```
  
+
+apply trac-13765-cyclo-embeddings.patch

[fchapoton]

comment:8

apply trac-13765-cyclo-embeddings.patch

[sagetrac-ehlen]

comment:9

• I tested the patch using Sage 5.5.
  • It is well documented and it solves the bug as far as I can see.
  • The documentation builds.

But I am not convinced that the implementation of NumberField_cyclotomic._log_gen should stay like it is. It is a private method but it might be used within the class for ther purposes later. Therefore, I think the output should always make sense. But it doesn't always and it is not completely consistent. Here is an Example.

This is what I expect from the documentation:

 sage: K.<zeta>=CyclotomicField(5, embedding=CC(exp(2*pi*i/5)))    
 sage: K._log_gen(10)
 sage: type(K._log_gen(10))
 <type 'NoneType'>

So, this is o.k., but:

 sage: K.<zeta>=CyclotomicField(5, embedding=CC(exp(2*pi*i/5)))    
 sage: K._log_gen(CDF(10))
 sage: 0
 sage: type(K._log_gen(CDF(10)))
 sage: <type 'sage.rings.integer.Integer'>

This behaviour is more or less documented in the docstring. It says that "If x is complex, returns the nearest such e whether or not the result is exact.". But I don't agree that it should be this way.

But I think this is not consistent and it should be fixed.