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Posted: **Mon Apr 11, 2022 5:56 am**

: Determine Q Q Where Q Is The Cubic Defined By The Polynomial 1 F X Y Z X3 2y3 423 Q X Y Z 2 F X Y Z 1 (24.61 KiB) Viewed 19 times

: Determine Q Q Where Q Is The Cubic Defined By The Polynomial 1 F X Y Z X3 2y3 423 Q X Y Z 2 F X Y Z 2 (5.72 KiB) Viewed 19 times

Determine Q(Q), where Q is the cubic defined by the polynomial: (1) F(X,Y,Z) = X3 + 2Y3 – 423 € Q[X,Y,Z). (2) F(X,Y,Z) = (Y + Z)3 - 2X3 € Q[X,Y,Z). 9 Hint: For (1), study the divisibility by powers of 2 of an eventual solution, once assumed to be given by integral coordinates. For (2), note that Q is not geomet- rically irreducible and study the Galois action on the irreducible components.

F(X, Y, Z) = X3 + 2Y3 – 423 € Q[X, Y, Z]

F(X, Y, Z) = (Y + 2)3 – 2X3 E Q[X, Y, Z].