miscellaneous propositions relating to injective system

英語は本当に使わないと忘れるなぁ。
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 $M_\lambda$ of the sequence is R-algebra and $f_{\lambda\mu},\ g_{\mu\lambda}$ are homomorphism over the R-algebra (for each corresponding subscript λ), then $\lim_\rightarrow M_\lambda,\ \lim_\leftarrow M_\lambda$ are R-algebra as well.

$1=(1_\lambda)_{\lambda\in\Lambda}\in \lim_\leftarrow M_\lambda$ and $\lim_\leftarrow M_\lambda$ is a subring of $\displaystyle{\prod_{\lambda\in\Lambda}M_\lambda}$ . Besides,

$\begin{array}{lcl} f_{\lambda\infty}(1_\lambda)&=&\pi\circ\tau_\lambda(1_\lambda) \\ &=& \pi((0,\ldots,0,1_\lambda,0,\ldots)) \\ &=& (0,\ldots,0,1_\lambda,0,\ldots) + R<\tau_\lambda(x)-\tau_\mu(f_{\lambda\mu}(x))>_{\lambda\leq\mu,\ x\in M_\lambda} \\ &=& f_{\mu\infty}\circ f_{\lambda\mu}(1_\lambda) \\ &=& f_{\mu\infty}(1_\mu) \end{array}$

becomes the identity element of $\lim_\rightarrow M_\lambda$ .

Proposition; for $m,\ n\in \mathbb{N}$

$M_{mn}=\big( \sum_{i=-\infty}^{-m} \sum_{j=n}^{\infty}\mathbb{C}\cdot X^{i+j} \big )\cap \mathbb{C}[X]\subset \mathbb{C}[X,1/X]$

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

When f is given as:

$f_{m,m+1}: X^{j-m}\mapsto X^{j-m-1}$

f provides an R-homomorphism from $M_{mn}$ to $M_{m+1,n}$ ; and n, similarly.

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

An application of the Galois Theory, for all $\tau$ in field L, conjugate of $\tau$ 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:

$\begin{array}{lcl} P &\Leftrightarrow & \exists n\in \mathbb{N}\cup \{0\}, 2^n40\text{ is an interior angle of a n-sided polygon }P_n \\ &\Leftrightarrow & P_{18}\text{ exists}\\ &\Leftrightarrow & \text{there exists intermediate field series of }Q(\zeta_{18})/Q\text{ that consist of}\\ && \text{quadratic extensions.} \end{array}$

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 $40,80,160=180(n-2)/n$ are reduced as n=18.