[with patch, positive review] new _latex_ and modified __repr__ for elements of relative number fields

[mabshoff]

Patch by Francis Clarke. He should fill in the details once his new trac account works.

Cheers,

Michael

[cremona]

This patch appliesw fine to 3.0.1 and the doctests in sage/rings/number_field all pass.

It looks fine to me.

[mabshoff]

Merged in Sage 3.0.2.alpha0

[fwclarke]

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| SAGE Version 3.0, Release Date: 2008-04-21                         |
| Type notebook() for the GUI, and license() for information.        |
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sage: A.<a> = CyclotomicField(3)
sage: P.<x> = PolynomialRing(A)
sage: B.<b> = NumberField(x^2 - 5)
sage: (a + b)^2
2*a*b - a + 4
sage: latex((a + b)^2)
\frac{4}{23} b^{3} - \frac{29}{23} b^{2} - \frac{8}{23} b + \frac{212}{23}

This is clearly wrong. What is happening is the element is being represented in the absolute number field, but using the variable appropriate to the relative field:

sage: C.<c> = B.absolute_field()
sage: from_abs, to_abs = C.structure()
sage: to_abs((a + b)^2)
4/23*c^3 - 29/23*c^2 - 8/23*c + 212/23
sage: latex(to_abs((a + b)^2))
\frac{4}{23} c^{3} - \frac{29}{23} c^{2} - \frac{8}{23} c + \frac{212}{23}

The patch below fixes this by providing a _latex_ function for elements of relative number fields.

It also simplifies __repr__ for such elements slightly.