{{{id=1|
ZZ.fraction_field()
///
Rational Field
}}}

{{{id=2|
R.<x>=QQ[]
///
}}}

{{{id=3|
R
///
Univariate Polynomial Ring in x over Rational Field
}}}

{{{id=4|
F=R.fraction_field()
///
}}}

{{{id=5|
F
///
Fraction Field of Univariate Polynomial Ring in x over Rational Field
}}}

{{{id=6|
F.random_element(degree=4)
///
(-x^4 + 2*x^3 - 1/2)/(-x^3 - x^2 + 1)
}}}

{{{id=8|
Q=QQ[sqrt(-3)]
///
}}}

{{{id=9|
Q
///
Number Field in a with defining polynomial x^2 + 3
}}}

{{{id=10|
two=Q(2)
///
}}}

{{{id=11|
two*two
///
4
}}}

{{{id=12|
another=Q((1-sqrt(-3)))*(Q(1+sqrt(-3)))
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_24.py", line 5, in <module>
    exec compile(ur'another=Q((_sage_const_1 -sqrt(-_sage_const_3 )))*(Q(_sage_const_1 +sqrt(-_sage_const_3 )))' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "parent.pyx", line 538, in sage.structure.parent.Parent.__call__ (sage/structure/parent.c:4956)
  File "coerce_maps.pyx", line 82, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (sage/structure/coerce_maps.c:3142)
  File "coerce_maps.pyx", line 77, in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (sage/structure/coerce_maps.c:3040)
  File "/usr/local/sage/local/lib/python2.6/site-packages/sage/rings/number_field/number_field.py", line 988, in _element_constructor_
    return self._coerce_non_number_field_element_in(x)
  File "/usr/local/sage/local/lib/python2.6/site-packages/sage/rings/number_field/number_field.py", line 4470, in _coerce_non_number_field_element_in
    raise TypeError, type(x)
TypeError: <type 'sage.symbolic.expression.Expression'>
}}}

{{{id=13|
T.<i,j,k>=QuaternionAlgebra(-1,-1)
///
}}}

{{{id=14|
T.is_noetherian??
///
}}}

{{{id=15|
P=PowerSeriesRing(QQ,'x',default_prec=50)
///
}}}

{{{id=20|
PowerSeriesRing?
///
}}}

{{{id=16|
num=P(x^4-3*x+2)
///
}}}

{{{id=17|
den=P(x^2+8*x-2)
///
}}}

{{{id=18|
num/den
///
-1 - 5/2*x - 21/2*x^2 - 173/4*x^3 - 715/4*x^4 - 5893/8*x^5 - 24287/8*x^6 - 200189/16*x^7 - 825043/16*x^8 - 6800533/32*x^9 - 28027175/32*x^10 - 231017933/64*x^11 - 952098907/64*x^12 - 7847809189/128*x^13 - 32343335663/128*x^14 - 266594494493/256*x^15 - 1098721313635/256*x^16 - 9056365003573/512*x^17 - 37324181327927/512*x^18 - 307649815626989/1024*x^19 - 1267923443835883/1024*x^20 - 10451037366314053/2048*x^21 - 43072072909092095/2048*x^22 - 355027620639050813/4096*x^23 - 1463182555465295347/4096*x^24 - 12060488064361413589/8192*x^25 - 49705134812910949703/8192*x^26 - 409701566567649011213/16384*x^27 - 1688511401083506994555/16384*x^28 - 13917792775235704967653/32768*x^29 - 57359682502026326865167/32768*x^30 - 472795252791446319888989/65536*x^31 - 1948540693667811606421123/65536*x^32 - 16061120802133939171257973/131072*x^33 - 66193023902203568291453015/131072*x^34 - 545605312019762485502882093/262144*x^35 - 2248614271981253510302981387/262144*x^36 - 18534519487869790567926733189/524288*x^37 - 76386692223460415782009914143/524288*x^38 - 629628057275553116824006046333/1048576*x^39 - 2594898921325672883078034099475/1048576*x^40 - 21388819427880936181448278842133/2097152*x^41 - 88150176632849417608871149468007/2097152*x^42 - 726590232490676277052417474586189/4194304*x^43 - 2994511106595554525818541047812763/4194304*x^44 - 24682679085255112483600745857088293/8388608*x^45 - 101725227447616004460221524476165935/8388608*x^46 - 838484498666183148165372941666415773/16777216*x^47 - 3455663222112348597121713291141829027/16777216*x^48 - 28483790275564971925139079270801047989/33554432*x^49 + O(x^50)
}}}

{{{id=19|

///
}}}