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Degree of Field Extensions and $F(\alpha)$ Suppose that $[F(\alpha):F]$ is odd. Show that $F(\alpha)=F(\alpha^{2})$. Generalize this statement if possible.    Clearly, we always have that $F(\alpha^{2}) \subset F(\alpha)$ by definition of the field generated by $F$ and $\alpha$ (assuming that we have a field extension $K/F$ where $\alpha \in K$). However, how can we show the other inclusion? If the degree is odd, then it follows that $\alpha$ will solve a polynomial of odd degree in $K[x]$ (for the coefficients in $F$). What would be a way to proceed from here? (and as well to generalize the statement?)  I was thinking of using the formula: $[F(\alpha):F]=[F(\alpha):F(\alpha^{2})][F(\alpha^{2}):F]$ and maybe proceed from there.   Thanks for the help. In the 70's or 80's I read a short story in which 2 astronauts looking for a nice place to eat lunch choose a spot at the pole of a small desolate planet, overlooking the world's only pool of surface water.  Due to the harsh extremes of heat, and lack of water, this is the home of the planet's single living organism, which has grown slowly over eons,  surviving the daily extremes, moving slowly over the bare rock, dissolving it with acidic digestive juices to get the nutrients it needs for survival. The men watch the organism while eating their lunch, then depart, unknowingly leaving death behind them in their garbage - cigarette butts, tin cans, wrappings - a bonanza that the creature moves over to dissolve and ingest, along with the human-based microscopic organisms that will destroy the alien being. The theme of the story is that although we reach out to the stars to seek life, we are much more connected to the lowest forms of life on Earth, than to any life we may find on other planets. I don't remember the author and title.  Can someone identify the story? Cannot Buy Land in Falkreath I'm describing a situation that happened in the past. To explain it, I want to use a description that is both true now and true when the situation happened.  Specifically, I want something like:     She touched me where my neck met my collarbone.   Since my neck is still attached to my collarbone (thankfully), I'm wondering if I shouldn't use the present tense here instead:     She touched me where my neck meets my collarbone.   Which is preferable, and why? A 398MB directory was only compressed to 393MB using 7Z and Normal ZIP compression. Is this normal? If so, why do people continue to use ZIP on Windows? How to delete GRUB entirely from GRUB rescue and boot Windows only Creating shapefile from current dataframe extents in layout view of ArcMap? Integral $\int\frac{dy}{(y^2 + ay + b)^{3/2}}$ Need the indefinite integral with $a,b$ constants. This is not a homework question. Need it for a physics problem. Another question on almost sure and convergence in probability Convergence in probability implies convergence on a subsequence almost surely.  But this means we fix a subsequence, such that $X_{n_k}$ converges for almost every $\omega$, right? The subsequence we pick does not depend on the $\omega$ right? How to pin Eclipse to the Unity launcher? Some irreducible character separates elements in different conjugacy classes Let $x$ and $y$ be elements that are not conjugate in $G$.  Then there is some irreducible character $\chi$ such that $\chi(x) \not = \chi(y)$.   Clearly the "irreducible" part isn't important, since any character can be written as the sum of irreducible characters, but I'm having trouble going beyond that.  I'd appreciate a good hint over a full answer, and I'd be most interested in a way to construct a group representation $\varphi:G \to GL(V)$ of $G$ such that the character of the representation takes different values on $x$ and $y$. apt-get update fails to fetch files, “Temporary failure resolving …” error Err http://archive.canonical.com natty InRelease     Err http://security.ubuntu.com oneiric-security InRelease                Err http://extras.ubuntu.com natty InRelease                             Err http://security.ubuntu.com oneiric-security Release.gpg   Temporary failure resolving ‘security.ubuntu.com’ Err http://archive.canonical.com natty Release.gpg   Temporary failure resolving ‘archive.canonical.com’ Err http://extras.ubuntu.com natty Release.gpg   Temporary failure resolving ‘extras.ubuntu.com’ Err http://gb.archive.ubuntu.com oneiric InRelease Err http://gb.archive.ubuntu.com oneiric-updates InRelease Err http://gb.archive.ubuntu.com natty-backports InRelease Err http://gb.archive.ubuntu.com oneiric Release.gpg   Temporary failure resolving ‘gb.archive.ubuntu.com’ Err http://gb.archive.ubuntu.com oneiric-updates Release.gpg   Temporary failure resolving ‘gb.archive.ubuntu.com’ Err http://gb.archive.ubuntu.com natty-backports Release.gpg   Temporary failure resolving ‘gb.archive.ubuntu.com’  Reading package lists... Done W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric/InRelease       W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric-updates/InRelease   W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/natty-backports/InRelease   W: Failed to fetch http://security.ubuntu.com/ubuntu/dists/oneiric-security/InRelease   W: Failed to fetch http://archive.canonical.com/ubuntu/dists/natty/InRelease   W: Failed to fetch http://extras.ubuntu.com/ubuntu/dists/natty/InRelease   W: Failed to fetch http://archive.canonical.com/ubuntu/dists/natty/Release.gpg  Temporary failure resolving ‘archive.canonical.com’ W: Failed to fetch http://security.ubuntu.com/ubuntu/dists/oneiric-security/Release.gpg  Temporary failure resolving ‘security.ubuntu.com’ W: Failed to fetch http://extras.ubuntu.com/ubuntu/dists/natty/Release.gpg  Temporary failure resolving ‘extras.ubuntu.com’ W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric/Release.gpg  Temporary failure resolving ‘gb.archive.ubuntu.com’ W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric-updates/Release.gpg  Temporary failure resolving ‘gb.archive.ubuntu.com’ W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/natty-backports/Release.gpg  Temporary failure resolving ‘gb.archive.ubuntu.com’ W: Some index files failed to download. They have been ignored, or old ones used instead.   This is what I'm seeing when I try to run sudo apt-get update. I did an update on my instance yesterday and am now experiencing this. Launch an app from within another (iPhone) Is it possible to launch any arbitrary iPhone application from within another app?, For example in my application if I want the user to push a button and launch right into the Phone app (close the current app, open the Phone app).  would this be possible? I know this can be done for making phone calls with the tel URL link, but I want to instead just have the Phone app launch without dialing any specific number. I would love to identify a sci-fi novel I read in the late 60s or 70s.  It followed a group of young space cadets through their training in the Space Academy.  The training culminated in small groups of cadets being sent through a gateway in the space station, into different worlds where they had to survive whatever conditions they met beyond the gateway, as part of their training.  One such trip went wrong and the group ended up stranded in a hostile world for days, hoping to be rescued by their trainers.  I can't remember much about the story or protagonists so this may be too vague, but I remember descriptions of a vast space station hall and these young cadets bringing whatever kit they hoped they might need - some brought breathing apparatus and full survival gear, but others came with very little as they believed they would not be sent into conditions where they could die without the right equipment.  They hadn't bargained for the power surge or whatever it was that misdirected them as they passed through the gateway!  I also remember rivalry between the cadets - I think the hero was male and quietly heading for top cadet of the year award... and I think at least one of the group died in the unknown planet before they were rescued.  Does this ring a bell with anyone?  I remember the mood and tension more than the details but it was a long time ago!  I have a 10-year old son who I think would love this kind of story.  If anyone can help you have my thanks! Oscillation of air particles and speed of sound wave A sound wave is essentially air particles oscillating parallel to the direction of travel of the wave.  We learnt that $v = f\lambda$, where $v$ is the speed of the wave, $f$ is the frequency of the wave and $\lambda$ is the wavelength of the wave.  Suppose the air particles oscillate with a frequency of $f_{particle}$. Is there any relationship between $f_{particle}$ and $v, f, \lambda$ of the entire sound wave?   I thought about this because intuitively, if the air particles vibrated faster, then the wave should travel faster, but I am unable to come up with any formula that describe this relationship (if it is even true!). Please explain the meaning of "otherwise than" in this sentence Please explain the meaning of the second part and explain "otherwise than" in part b     a person’s UK property business consists of:      (a) every business which the person carries on for generating income from land in the United Kingdom; and      (b) every transaction which the person enters into for that purpose otherwise than in the course of such a business. Commutative non Noetherian rings in which all maximal ideals are finitely generated In commutative rings we have the following  Theorem. $R$ is Noetherian if and only if each prime ideal of $R$ is finitely generated.  From this Theorem I am looking for commutative rings $R$ in which every maximal ideal is finitely generated but $R$ is non Noetherian.  Question: Is there a straightforward example of a commutative ring $R$ so that each maximal ideal is finitely generated, but $R$ is non Noetherian?  Thank You Let $k$ be a finite field of order $q$ in characteristic $p$, let $n$ be a positive integer not divisible by $p$, and let $K$ be the splitting field of $X^n-1$ over $k$. Prove that $[K:k]$ equals the smallest positive integer $d$ such that $n\mid q^d-1$.  My approach: It's easy to show that if $k$ is a finite field of order $q$ with $\text{char} \ k=p$ then $k=\mathbb{F}_{q}$ with $q=p^m$ with $m\geq 1$. If $K$ is the splitting field of $X^n-1$ over $k$ then $K=\mathbb{F}_{q^r}$ for some suitable $r\geq 1$.  Let $\alpha \in K=\mathbb{F}_{q^r}$ be a root of $X^n-1$ then $\alpha^n-1=0$ and $\alpha\neq 0$.   Thus, $\alpha^n=1$ and $\alpha \in \mathbb{F}^{\times}_{q^r}$ where $\mathbb{F}^{\times}_{q^r}$ is the cyclic group of order $q^r-1$.  But I have some difficulties:  1) From $\alpha^n=1$ it follows that $o(\alpha)\mid n$. How to show that $o(\alpha)=n$? I was trying to show it using that $(n,p)=1$ but I failed.  Remark: Because if we can show that $o(\alpha)=n$ then the desired result follows immediately from cyclicity of multiplicative group of field $\mathbb{F}_{q^r}$.  I have spent one day in order to come up with this approach. So please do not duplicate this question and can anyone show how to answer my question, please.  Detailed explanation would be great! What is the current state of apple device (iPhone, iPod, etc.) support? Orthogonal basis for infinite-dimensional vector spaces