### checking if element of QQbar is in QQ

I would like to know the most efficient way of checking whether a given element `x`

which is in `QQbar`

actually is in `QQ`

. The obvious way, that is asking `x in QQ`

, does not always work: I have an example for which no answer is produced in several minutes, while `x.degree()`

returns 18 very rapidly (so the element is certainly not in `QQ`

).

Now I check `x.degree() == 1`

, which is better, but on some examples it's still impossible to get a result in a reasonable time.

So I was wondering if anyone knew the very fastest way of obtaining the answer. (I have hundreds of thousands of elements of `QQbar`

, and I need to know which of them are in `QQ`

...)

*Edit* : I have tried `sage_input(x)`

on one number, for which `x.degree()`

seemed to take a very long time, and it does seem huge indeed, so perhaps the slowness is to be expected. It starts with

```
R.<y> = QQ[]
RIF512 = RealIntervalField(512)
RR512 = RealField(512)
v = QQbar.polynomial_root(AA.common_polynomial(y^108 + 68004*y^107 + 2292971238*y^106 + 51109008051132*y^105 + 847128116941893513*y^104 + 11136459321510129312480*y^103 + 120944040275150561015208000*y^102 + 1115986520588898954106738693008*y^101 + 8930751629479221489446680712620452*y^100 + 62961183935900437502097267939514726608*y^99 + 395885483512322567640638781252627560094624*y^98 + 2242347971328556178478595187470182159924789216*y^97 + 11535503038920918187810269479084854355202789445188*y^96 + 54269390660779471643106858668214098741160413790240672*y^95 + 234851336491050360542100170120549418461242139278483722168*y^94 + 939569003630299986203466023916881930753077854558405425462160*y^93 + 3490211835252670149247256331453220478710134468215140726060796886*y^92 + 12084161135860812171548648158452586929299853124159455422191690733976*y^91 + 39127445477819813897097350578201509138815301746365355947191046849364460*y^90 + 118834193635640440426978357924069602204303569995237870942404129179353253784*y^89 + 339431414930153584624175760167927030458787824395559124832063620982593184272454*y^88 + 914008587316979610169324522266393486470316485890810584833535325834539591845375520*y^87 + 2325255525666058595638903828649302942489341130326923157478734432098334031117961245944*y^86 + 5599623716538838373608322537699558868191795870505495950293442445752147110596713205528240*y^85 + 12787408442724754213893995626529393993223307270863826470321168842894295344087154710931173436*y^84 + 27735823920352260774281783498610617352976434668417400157621968451890240290776126525793240201904*y^83 + 57223330240259045601668613415140173457018450187726192871526041989829984449684813838150321898877344*y^82 + 112451025217727515917262506500621531971063596471347132306074496600842842754197965290920211733860360672*y^81 + 210740669967907045884992444493112155229328186358263361745499064895003334115907302697598496922153739227836*y^80 +
```

and goes on like that for so many hundreds of lines that I can't even count.