Find the splitting field for f(x)=(x^2+x+2)(x^2+2x+2)over Z_3.write f(x) as a product of linear factors?

Since x^2 + 2x +2 is irreducible over Z_3 (x = 0,1, and 2 are not solutions to f(x) AND f(x) is quadratic), let's look at the field extension Z_3(t), where t is a root of x^2 -2x + 2.





By long division, x^2 + 2x + 2 = (x -t) (x - (t+2)). (Note that the ';remiander'; is t^2 -2t +2, which is 0 by definition of t.)





Since t+2 is in Z_3(t), this extension field is the splitting field of f(x) over Z_3 and we have written f(x) as the product of linear factors.





Note: We can construct Z_3(t) explicitly via the isomorphism


Z_3(t) = Z_3[x]/(f(x)), giving us a field with only 9 elements!Find the splitting field for f(x)=(x^2+x+2)(x^2+2x+2)over Z_3.write f(x) as a product of linear factors?
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