R.<a,b,c> = QQ[] what is that?

While playing with Jaap's wish, I finally got fedup and decided to study what the funny syntax "R.<a,b,c> = QQ[]" really means. So I did:

sage: R. = QQ[]
sage: preparse("R. = QQ[]")
"R = QQ['a, b, c']; (a, b, c,) = R._first_ngens(Integer(3))"

Now everything is crystal clear! Let's study what QQ is:

sage: QQ?   
Type:  RationalField
Base Class: 
String Form: Rational Field
Namespace: Interactive
Docstring:
    
        The class class{RationalField} represents the field Q of
        rational numbers.

Cool, why not. So what is the _first_ngens method doing? Let's find out:

sage: R._first_ngens?
Type:  builtin_function_or_method
Base Class: 
String Form: 
Namespace: Interactive

Hm, not really useful. Let's push harder:

sage: R._first_ngens??
Type:  builtin_function_or_method
Base Class: 
String Form: 
Namespace: Interactive
Source:
    def _first_ngens(self, n):
        v = self.gens()
        return v[:n]

So the equivalent code is this:

sage: R.gens()
(a, b, c)
sage: R.gens()[:3]
(a, b, c)

Cool, why not. So what is the R.gens() doing?

sage: R.gens? 
Type:  builtin_function_or_method
Base Class: 
String Form: 
Namespace: Interactive
Docstring:
    
            Return the tuple of variables in self.
    
            EXAMPLES:
                sage: P. = QQ[]
                sage: P.gens()
                (x, y, z)
    
                sage: P = MPolynomialRing(QQ,10,'x')
                sage: P.gens()
                (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)
    
                sage: P. = MPolynomialRing(QQ,2) # weird names
                sage: P.gens()
                (SAGE, SINGULAR)

Ah, that's the answer we want. :)