If I have two algebraic numbers &#x03B1;<!-- α --> and &#x03B2;<!-- β --> and a rational fun

Antoine Hill

Antoine Hill

Answered question

2022-05-24

If I have two algebraic numbers α and β and a rational function w with rational coefficients (a function that's the ratio of two rational polynomials) that relates the two α = w ( β ) if I were to substitute β with one of it's algebraic cojugates β in the rational function will I get a conjugate of α or rather is w ( β ) an algebraic conjugate of alpha? I feel like there is a very obvious counter example but I've been struggling to find one.

Answer & Explanation

szilincsifs

szilincsifs

Beginner2022-05-25Added 15 answers

Let f , F Z [ X ] be irreducible. Let g , h Z [ X ] with h 0. Let β C with f ( β ) = 0 and F ( g ( β ) h ( β ) ) = 0 (and in particular, h ( β ) 0). Note that by cancellation of h`s,
p ( X ) := h ( X ) deg F F ( g ( X ) h ( X ) )
is actually Z [ X ]. As p ( β ) = 0, we conclude that β's minimal polynomial f divides p. On the other hand, h ( β ) 0 implies that f does not divide (or even has a common factor with) p. But that means that for every β with f ( β ) = 0, we have p ( β ) = 0 and h ( β ) 0, hence
F ( g ( β ) h ( β ) ) = 0 ,
i.e., g ( β ) h ( β ) is a conjugate of g ( β ) h ( β ) .

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