# Real Algebraic Scheme question

I apologize if this question is too naive.

I need to know the irreducible components of an algebraic scheme defined over $\mathbb{R}$. I can get Sage to do this if I consider the scheme is defined over $\mathbb{Q}$, but this is not sufficient to answer my question over $\mathbb{R}$.

Can Sage actually do this for real algebraic schemes?

and here is the code I tried:

K = RealField()
A9 = AffineSpace(K, 2, 'a,b')
A9.coordinate_ring().inject_variables()
W=A9.subscheme([a*b^2]);
W.is_irreducible()

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As a general advice, note that when you write

sage: K = RealField()


You do not define the genuine real field, but an approximation made of floating-points numbers with 53 bits of precision:

sage: K
Real Field with 53 bits of precision


This is not a safe place for doing algebraic geometry. Usually, i would recommend to work on the algebraic field QQbar or the algebraic real field AA.

Unfortunately, regarding your question, defining K as AA does not solve the problem since the Singular library which is used under the hood for your computation is not able to deal with it.

Sorry not to be able to help more.

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