Pushout for real embedded number fields

[videlec]

Implement the following pushout

sage: K.<a> = NumberField(x^2 - 3, embedding=AA(3)**(1/2))
sage: L.<b> = NumberField(x^2 - 2, embedding=AA(2)**(1/2))
sage: a+b
3.146264369941973?
sage: parent(_)
Algebraic Real Field
sage: a < b
False

CC: @jdemeyer @tscrim

Component: number fields

Keywords: days74, days84

Author: Vincent Delecroix

Branch/Commit: 4a52745

Reviewer: Jean-Philippe Labbé

Issue created by migration from https://trac.sagemath.org/ticket/20746

[videlec]

Branch: u/vdelecroix/20746

[videlec]

Commit: 4a52745

[videlec]

New commits:

4a52745

Trac 20746: pushout for real embedded number fields

[pjbruin]

comment:4

Isn't it a bit drastic to declare AA to be the pushout rather than to create a suitable number field? I think a better solution would be to use the existing composite_fields() method, which respects embeddings:

--- a/src/sage/rings/number_field/number_field_base.pyx
+++ b/src/sage/rings/number_field/number_field_base.pyx
@@ -350,3 +350,7 @@ cdef class NumberField(Field):
         else:
             raise ValueError("No embedding set. You need to specify a a real embedding.")
 
+    def _pushout_(self, other):
+        if (isinstance(other, NumberField) and self._embedded_real and
+            (<NumberField>other)._embedded_real):
+            return self.composite_fields(other)[0]

This makes the examples in the current branch work just as well (you can see the numerical values after coercing into AA).

[videlec]

comment:5

Fair enough. Though it is more expensive.

[videlec]

comment:6

And this is order dependent

sage: K.<a> = NumberField(x^2 - 3, embedding=AA(3)**(1/2))
sage: L.<b> = NumberField(x^2 - 2, embedding=AA(2)**(1/2))
sage: a+b
ab^3 - 10*ab

versus

sage: K.<a> = NumberField(x^2 - 3, embedding=AA(3)**(1/2))
sage: L.<b> = NumberField(x^2 - 2, embedding=AA(2)**(1/2))
sage: b+a
-ba^3 + 10*ba

[jplab]

comment:8

This coercion has its advantages:

• makes it possible to do arithmetic operations with elements with two different NumberFields as parents. Right now returns a TypeError:

    sage: J = NumberField(x^2 - 2,'s')
    sage: s = J.gens()[0]
    sage: K = NumberField(x^3 - 2,'t')
    sage: t = K.gens()[0]
    sage: s * t
    Traceback (most recent call last):
    ...
    TypeError: unsupported operand parent(s) for *: ...

• It is lazy, it does have to compute the join.

Depending on the usage, one may want to get the join of the fields using composite_field, this introduces another variable and is order dependant. I guess that if one wants to do computations on the long run with that field, one would already use the right field from start.

IMHO, this ticket already fixes an operation that one would naturally want to have at hand. So, I would make this a positive review.

[jplab]

Reviewer: Jean-Philippe Labbé

[jplab]

Changed keywords from days74 to days74, days84

[vbraun]

Changed branch from u/vdelecroix/20746 to 4a52745