Let F be an extension field of K. Let a ∈ F be algebraic over K, and let t ∈ F be transcendental over K. Show that a+t is transcendental over K. I am completely lost. Can anyone help at all? Thanks!
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Use a proof by contradiction. Let x= a+ t. If x is NOT transcendental, it satisfies some polynomial equation, P(x)= 0, with coefficients in K. Then P(a+ t)= 0.
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