Let E and E' be two extensions of F in ‘...'hich a polynomial f over F splits. Prove that there exists an isomorphism (1) from the splitting field of f in E to the splitting field of f in E such that...

Let E and E' be two extensions of F in ‘...'hich a polynomial f over F splits. Prove that there exists an isomorphism (1) from the splitting field of f in E to the splitting field of f in E such that cp(c) = c for every c E F. 2.) Find the splitting field of x4 + x2 + 1 = (x2 + x + 1)(x2 — x + 1) over Q. 3.) Let E be an extension field of F and let a E E be an algebraic element over F. Show that F(a) = F[x] / where f is a minimal polynomial of a over F. 4.) Prove that two finite fields with same number of elements are isomorphic. 5.) Prove that every algebraic extension of a prefect field is perfect.




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