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 2.2.12 A Groebner Basis Example
A Groebner basis of an ideal I is calculated with the command GBasis(I), as illustrated in the following example.

Let r be a root of the equation x^7-x-1 over the rationals. The minimal polynomial of (4r-1)/r^3 can be found by computing the reduced Groebner basis of the ideal (x^7-x-1,x^3y-4x+1) with respect to the lexicographic term-ordering with x>y.

 Example
 ``` Use R ::= QQ[x,y], Lex; Set Indentation; -- to improve the appearance of the output G := GBasis(Ideal(x^7-x-1,x^3y-4x+1)); G; [ 1602818757152090759440/34524608236181199361x - 4457540/5875764481y^7 - 47746460716124220/34524608236181199361y^6 + 890175715271333840/34524608236181199361y^5 - 3541992534667352220/34524608236181199361y^4 - 55943894513139464160/34524608236181199361y^3 - 56473654361333280980/34524608236181199361y^2 - 27971979712025453040/34524608236181199361y - 400704689288022689860/34524608236181199361, 1/16384y^7 - 5/16384y^6 + 147/16384y^4 + 5/128y^3 - 31/16384y^2 + 17/128y - 20479/16384] ------------------------------- Len(G); 2 ------------------------------- F := 16384*G; -- clear denominators F; y^7 - 5y^6 + 147y^4 + 640y^3 - 31y^2 + 2176y - 20479 ------------------------------- ```
The Groebner basis is reported as a list with two elements. The second gives a univariate polynomial which is the minimal polynomial for r.

Note that the statement declaring the ring includes the modifier, Lex. Without this modifier, the default term-ordering, DegRevLex, is used. The command Set Indentation forces each polynomial of the Groebner basis to be printed on a new line.