I am trying to prove ex 2.13 from Jech's book on set theory:     Ex 2.13: A limit ordinal $\gamma$ is indecomposable if and only if $\alpha + \gamma = \gamma$ if and only if $\gamma = \omega^{\alpha}$ for some $\alpha$.   I think this is equivalent to proving that the 3 following statement are equivalent:   $\gamma$ is indecomposable (i.e. there are no $\alpha,\beta &lt; \gamma$ s.t. $\gamma = \alpha + \beta$). For every $\alpha &lt; \gamma$, $\alpha + \gamma = \gamma$. There exists $\alpha$ s.t. $\gamma = \omega^{\alpha}$.   I think that I can prove $(1) \rightarrow (3)$ since it seems almost straight forward from Cantor's normal form theorem. I can also prove that $(1) \Rightarrow (2)$ since, for every $\alpha &lt; \gamma$, there exists a unigue $\delta$ s.t. $\alpha + \delta = \gamma$. Given, (1), $\delta \geq \gamma$. But $\delta &gt; \gamma \Rightarrow \alpha + \delta &gt; \gamma$ and we are left with $\delta = \gamma$.   My problem is with the other directions. Any help?
A limit ordinal $\gamma&gt;0$ is said to be indecomposable iff $\nexists\alpha,\beta&lt;\gamma$ such that $\alpha+\beta=\gamma$.  In view of this definition, I’m trying to prove the equivalence of the following statements for limit ordinals $\gamma$:  (1) $\gamma$ is indecomposable.  (2) $\alpha+\gamma=\gamma$ $\forall\alpha&lt;\gamma$.  (3) $\gamma=\omega^\alpha$ for some $\alpha\in\mathrm{ON}$.  I’ve been able to get $(1)\implies(2)$ and $(3)\implies(1)$, but I have nothing more than scratch work for $(2)\implies(3)$.  Any suggestions would be appreciated.
as of 4/2015 OS X Kindle App: Where are the books located? After I updated my Kindle for Mac app 2 mos. ago I can no longer find the azw book files on my Mac to convert them so I can read them on my KOBO ereader. I use to find them in a My Kindle Content file but that has disappeared. I have Ult. DRM &amp; Calibre but they won't work without the azw files, and I can't find them on my mac anymore. Can anyone help me? The answers on this site about this are years old &amp; do not apply to my problem. Page 105 of D. Burton's A First Course in Rings and Ideals reads     It is not difficult to show that if (I'll call them $n$ and $m$ instead of $n_1$ and $n_2$) $n$, $m$ are square free integers, then $\mathbb{Q}(\sqrt{n}) \cong \mathbb{Q}(\sqrt{m})$ if and only if $n=m$.   Well, it is getting difficult for me, any help or would be appreciated.    My progress so far: Suppose $\phi :\mathbb{Q}(\sqrt{n})\to\mathbb{Q}(\sqrt{m})$ is an isomorphism. Then $\phi(u)=u$ for every $u \in \mathbb{Q}$, so that $\phi(\sqrt{n})^2=\phi(n)=n$. If $\phi(\sqrt{n})=a+b\sqrt{m}$ for some $a,b\in\mathbb{Q}$, then $(a+b\sqrt{m})^2=a^2+b^2m + 2ab\sqrt{m}=n$ implies $$a^2+b^2m = n$$ and $$2ab=0.$$  Then $b=0$ cannot happen since that would imply that $\sqrt{n}\in\mathbb{Q}$, where $n$ is a square-free integer. Also $a=b=0$ cannot happen. Hence $a=0$ and we have $$b^2m=n.$$ If $\psi:\mathbb{Q}(\sqrt{n})\to\mathbb{Q}(\sqrt{m})$ were an isomorphism then there would exist some $s\in\mathbb{Q}$ such that $$s^2m=n,$$ then $s=b$ or $s=-b$. Therefore the only isomorphisms from $\mathbb{Q}(\sqrt{n})$ to $\mathbb{Q}(\sqrt{n})$ are $\phi(u+v\sqrt{n})=u+vs\sqrt{m}$ and $\psi(u+v\sqrt{n})=u-vs\sqrt{m}$.    I don't know if there's an easier way and I don't know if I'm in the correct way here. Thanks beforehand :) How to use an optimization solver to get t-stats and p-values for the estimates? can't get placeholder attribute to work in IE using iframe $\mathbb{R}P^4$ and $\mathbb{R}P^6$ do not admit fields of tangent $2$-planes See the related question . This is the second part of question 4-C in Milnor and Stasheff's book on characteristic classes. In the solution to the first part, we rely on the fact that having a field of tangent $1$-planes allows us to split the tangent bundle into a direct sum. It doesn't seem like this should be true for a field of tangent $2$-planes (otherwise the question wouldn't just be asking about $P^4$ and $P^6$), so I'm not sure exactly where to start.  edit: actually, since $P^n$ can be given a Riemannian metric, any subbundle should be a Whitney summand; this doesn't answer the question as to why they only ask for $P^4$ and $P^6$ though, so I'm still confused. 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Then $d|a$ and $d|b$ then $d^2|a^2$ and $d^2|b^2$. i.e $d^2$ divides $a^2 ~~\&amp;~~ b^2$. Unmangling the result of std::type_info::name Rewrite Sharepoint URL (Sharepoint 2013) I'd like to rewrite all my site urls in the way the the user can't figure out the exact name of the the page his viewing. let's say that my site is "www.justarandomsite.com"  and may page is "page.aspx" I' like to have instead of "www.justarandomsite.com/pages/page.aspx" sothing like this: "www.justarandomsite.com/skdhfjksdgfjksdgfjdsgvdjhsgfjdsdsjfgkdsjgfkdsgfjdsgfjkdsg" Explicit vs implicit SQL joins Can I manually put an external hard disk to sleep? Detecting unused packages in LaTeX document I have quite a bad habit of copy-and-pasting my packages from one LaTeX document to another. Is there any way we can detect unused packages (i.e. \usepackage commands) so that they can be removed from the LaTeX document? Visiting Greece with back-to-back Schengen visas (One from Germany, other from Italy) I have a production server running with the following flag: -XX:+HeapDumpOnOutOfMemoryError  Last night it generated a java-38942.hprof file when our server encountered a heap error. It turns out that the developers of the system knew of the flag but no way to get any useful information from it.  Any ideas? How to monitor who is pinging me? Is there a way to know who is pinging my PC on my network? Is it possible to monitor all ICMP packets? How can I do that? Will doing timelapses break my camera? I was planning to do some timelapses in the near future, until I realised that if I leave my camera taking pictures for a week or 2, I will soon reach the max number of shutter count and my camera would break.   Then I read from an unverified source that during timelapse the shutter doesn't move, so there is no way to break it. I am confused.   Is it possible to take timelapses without physically destroying the shutter in a month? In particular, I am interested in Canon 80D model. How precise is the frequency of the AC electricity network? If I make a digital clock which first detects the AC frequency roughly (whether it is 50, 60, 100Hz, etc), then uses it as its clock pulse source, how precise would it be? Would it work world wide precisely?  For instance, in USA, what is the mean and standard deviation of the 60Hz AC frequency? Number of normal subgroups of a non abelian group of order $21$... CSIR December $2013$ Question is to :  Find the number of Normal subgroups of a nonabelian group $G$ of order $21$ other than $\{e\}$ and $G$.  What I have done so far is :  As $|G|=21=3\cdot7$  we have :  No. of sylow $3$ subgroups $1+3k$ dividing $7$ leaving out possibilities $1$ or $7$  No. of sylow $7$ subgroups $1+7k$ dividing $3$ leaving out only possibilities $1$.  So, we have a unique sylow $7$ subgroup and so it is normal.  I remember somehow that any normal group should come from a normal sylow subgroup or something like that.  S0, I prefer to conclude there is only one Normal subgroup for a non abelian group of order $21$.  Please let me know if this is true and please help me to fill that gap :      I remember somehow that any normal group should come from a normal sylow subgroup or something like that.   Thank you.