Cryptography Reference
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roots. In particular, σ ( r )= y for some r . Therefore, y is in the image of σ .
This proves that σ is surjective, so σ is an automorphism of C .
Since
σ j ( α )= α + j,
the set of images of α under automorphisms of C is infinite, in contradiction
to our assumption. Therefore, α cannot be transcendental, hence must be
algebraic.
REMARK C.8 In Proposition C.7, the assumption that the set is finite
can be changed to assuming that the set is countable, with essentially the
same proof. Namely, if α is transcendental, then, for any γ ∈ S ,thereisan
automorphism σ satisfying σ ( α )= α + γ . The fact that S is uncountable
yields the result.
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