miscellaneous propositions relating to injective system


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.

The important assertion is used for the second arrow:

    \[Q(\zeta_{18})/Q\cong (Z/18Z)^\times\]

Let G be {\rm Gal}(Q(\zeta_{18})/Q), we have 2 possible Galois group extension series as follows:

    \[\begin{array}{l} 1\subset \{1,7,13\} \subset \{1,5,7,11,13,17\}=G \\ 1\subset \{1,17\} \subset \{1,5,7,11,13,17\}=G \end{array}\]

which indicate that both contain that of 3 at extension degree.
Concluded negatively, non.