How do I compute the reduced Groebner Basis?

I am trying to do a question on Hilbert's Weak Nullenstratz theorem for the 3 colouring of vertices and i know i need to compute the reduced Groebner Basis (GB) for the following Ideal:

$$I:= <x_{1}^{3}-1, x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}, x_{1}^{2}+x_{1}x_{3}+x_{3}^{2}, x_{2}^{3}-1, x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}, x_{3}^{3}-1>$$

Using maple i know that the reduced GB is $$I:= <x_{1}+x_{2}+x_{3}, x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}, x_{3}^{3}-1 >$$ However i am struggling to get to this when finding it for myself. When i have the S-Poly of the first two equations i notice that its the same as the 4th equation so can i remove the 4th equation?

Can someone explain how i get to the reduced GB?

The reduced Gröbner basis still depends on the monomial order chosen. I obtained $$\mathcal{G}=\{x_1^3-1,\,x_2^2 + x_2x_1 + x_1^2,x_3 + x_2 + x_1\},$$ by using the Buchberger algorithm, doing multivariate division.
• sorry should have added i am using the lex order $x_{1}> x_{2}> x_{3}$, guess your method was with the lex in reverse? – user496388 Apr 6 '18 at 15:45