# miscellaneous propositions relating to injective system

40度はコンパスと定規で作図できない！ことの証明など。

Proposition; of inductive or direct system, each factor of the sequence is R-algebra and are homomorphism over the R-algebra (for each corresponding subscript λ), then are R-algebra as well. and is a subring of . Besides, becomes the identity element of .

Proposition; for  , as of m, this system interprets inductive; and of n direct system.

When f is given as: f provides an R-homomorphism from to ; and n, similarly.

Proposition; is angle of 40 drawable with compasses and ruler?

An application of the Galois Theory, for all in field L, conjugate of over the base field K belongs to L, then L/K is a regular extension.
Presuming the separability of extension over perfect field K(:=Q) is equivalent to having L/K an algebraic extension.

Here the equivalency of the proposition P denoted as below: The first arrow implies the property of a drawable angle; such that only the power of 2 division is allowed operation if it is drawable, and only when it appears as an interior angle of polygon.
We immediately figure are reduced as n=18.

The important assertion is used for the second arrow: Let G be , we have 2 possible Galois group extension series as follows: which indicate that both contain that of 3 at extension degree.
Concluded negatively, non.

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