Make SymmetricFunctions work with define_implicitly

[mantepse]

Problem Description

The following should (probably) work after #37033, but doesn't:

sage: h = SymmetricFunctions(QQ).h()
sage: L.<t, u> = LazyPowerSeriesRing(h)
sage: D = L.undefined()
sage: s1 = L.sum(lambda n: h[n]*t^(n+1)*u^(n-1), 1)
sage: L.define_implicitly([D], [u*D - u - u*s1*D - t*(D - D(t, 0))])
sage: D

This involves various shortcomings of the ring of symmetric functions, in particular, that it does not come with a fraction field, see #34347

Proposed Solution

implement the missing methods

Alternatives Considered

live without.

Additional Information

No response

Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.