## Find the conjuages of a given algebraic element over QQ

Find the minimal polynomial and conjugates of each of the following elements over : , 4, , and .

Note that has as a root. is Eisenstein at 3, hence irreducible over . Thus is the minimal polynomial of over . The conjugates of are .

Note that has 4 as a root. Since is linear, it is irreducible, and so is the minimal polynomial of 4 over . The only conjugate of 4 is 4 itself.

Note that has as a root. is Eisenstein at 3, hence irreducible over . Thus is the minimal polynomial of . The conjugates of are .

Note that has as a root. is Eisenstein at 5, hence irreducible over . The conjugates of are , , and .

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