Testing ideals for primality ( examples over k = \342\204\232 )F := [x^2+y+z,x+y^2+z,x+y+z^2];G := Groebner[Basis](F,plex(x,y,z));factor(G[1]);Since G[1] factors over \342\204\232 this means I is not prime over \342\204\232.F := [x^2+1,y^2+1,z^2+1];Groebner[Basis](F,plex(x,y,z));Notice that F is a Groebner basis for 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 wrt. any monomial ordering by Proposition 4 of 2.9 since the leading monomials of the generators are 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 for any monomial ordering. Let's try a linear transformation on the ideal I of the form y \342\206\222 y + a x, z \342\206\222 z + b x + c y.Ft := subs(y=y+3*x,z=z+5*x,F);Gt := Groebner[Basis](Ft,plex(x,y,z));factor(Gt[1]);Ft := subs(y=y+3*x,z=z+5*x-3*y,F);Gt := Groebner[Basis](Ft,plex(x,y,z));f := Gt[1];The Groebner basis is of the form 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 for some constants 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. Thus I is prime \342\207\224JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInpGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is irreducible over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NDc0O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Also I is radical \342\207\224LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInpGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is square-free (i.e., has no repeated factors).factor(f);Hence we conclude I is radical but it is not prime over \342\204\232.