### Intersection of polynomial Ideals over $\mathbb{R}$

I am trying to compute the intersection of Ideals over $\mathbb{R}[x,y]$, but I get problems from the coefficient $\frac{1}{\sqrt{2}}$. This is my code:

R.<x,y>=PolynomialRing(RR,order='lex')

I=Ideal([(x^2+y^2-1),(x*y),(y^3-y)])

I5=Ideal([x-1/sqrt(2),y-1/sqrt(2)])

I6=Ideal([x+1/sqrt(2),y-1/sqrt(2)])

I7=Ideal([x+1/sqrt(2),y+1/sqrt(2)])

I8=Ideal([x-1/sqrt(2),y+1/sqrt(2)])

J=I.intersection(I5,I6,I7,I8)

and this is the error I get:

TypeError: Intersection is only available for ideals of the same ring.

So when I ask if

I5 in R

the answer is False. I also tried with QQbar but same result, can someone explain this?
Thanks!

P.S. Sorry I can't seem to Blockquote, if some mod does it would be nice