The question is:  Let L / K be a finite (not necessarily Galois) extension of algebraic number fields and N / K the normal closure of L / K. Show that a prime ideal p of K is totally split in L if and only if it is totally split in N. Hint: Use the double coset decomposition H\G/GP, where G = G(N/K), H = G(N/L) and GP , is the decomposition group of a prime ideal P over p.  My question is how to use this hint to solve this problem; please give me some advice.
I'm reading Algebraic Number Theory by Jurgen Neukirch. I have some problems with some of the exercises in Section 9 of Chapter 1.  They are:  1) If $L / K$ is a Galois extension of algebraic number fields with non-cyclic Galois group, then there are at most finitely many non-split prime ideals of $K$.  2) Let $L / K$ be a finite (not necessarily Galois) extension of algebraic number fields and $N / K$ the normal closure of $L / K$. Show that a prime ideal $p$ of $K$ is totally split in $L$ if and only if it is totally split in $N$.  I have worked on them for a long time but couldn't get any idea of it.   Can you please help? Thank you!
Hyperref: Changing the linkcolor locally in the TOC I found that the link-color for TOC text is identical to the link-color for Figures and Tables. When I change the color to black in the TOC also the color for Figures/Tables citation etc. is changed.  Is there a way to separately set the colors inside the TOC and outside? What was Stack Exchange 1.0? I've seen a few users mention that certain SE sites like  and  are unique, as they were originally created as Stack Exchange 1.0 sites.  As a fairly recent user, I don't know what Stack Exchange 1.0 was like. The current version of Stack Exchange is the only version I've ever known.   What was Stack Exchange 1.0? What changed between SE 1.0 and 2.0? From time to time I lose all edits because of undo in Object Mode. If I enter immediately after Undo in Edit Mode I can't redo, and switching again in Object results in losing all edit history.  How can I resolve this? Separating undo history by mode or linear/unified will both be acceptable.  Edit:I think the most preferable solution will be to disable/skip edit undos when in object mode through an addon. Geometric interpretation of Electromagnetism What's the difference between an argument and a parameter? When verbally talking about methods, I'm never sure whether to use the word argument or parameter or something else. Either way the other people know what I mean, but what's correct, and what's the history of the terms?  I'm a C# programmer, but I also wonder whether people use different terms in different languages.  For the record I'm self-taught without a background in Computer Science. (Please don't tell me to read  because I'm asking this for the benefit of other people who don't already have a copy of 's marvellous book.)  Summary  The general consensus seems to be that it's OK to use these terms interchangeably in a team environment. Except perhaps when you're defining the precise terminology; then you can also use "formal argument/parameter" and "actual argument/parameter" to disambiguate. Cat 5/6 in same conduit as power: is NM-B required, or is THHN acceptable? Preface: I'd like to avoid the question of whether one should run data and power together; my question is focused on Code compliance.  I'm trying to ensure I interpret Code correctly when running Cat 5/6 cabling and power together in the same conduit.  For starters, I understand that nothing in Code prohibits one from doing this; I'm trying to ensure I understand how to do it properly.  I'm wiring a basement office for power and data, and I'll be running the electrical in EMT.  My plan was to pull THHN for the power rather than pull NM-B, but then 800.133(2) Exception 1 made me scratch my head.  Is my understanding of the Article that if I want to run both data (Cat6) and power in the same conduit, that the power needs to be sheathed (i.e. NM-B).  Is it unacceptable to pull individual THHN conductors and the Cat6 together in the same conduit?     800.133 2) Other Applications. Communications wires and cables shall be separated at least 50 mm (2 in.) from conductors of any electric   light, power, Class 1, non–power-limited fire alarm, or medium-power   network-powered broadband communications circuits.      Exception No. 1: Where either (1) all of the conductors of the   electric light, power, Class 1, non–power-limited fire alarm, and   medium-power network-powered broadband communications circuits are in   a raceway or in metal-sheathed, metal-clad, nonmetallic-sheathed, Type   AC, or Type UF cables, or (2) all of the conductors of communications   circuits are encased in raceway. Do I have to peel red potatoes before baking them? I recently made some potato wedges in the oven. I shared this recipe with my relative, who admonished me for not properly peeling the red potatoes before baking them.  She claims that there are disease and viruses that lie on the skin, so peeling it will get rid of it.  I bake it, so I assume that even the high heat in the oven would kill bacteria and germs. And besides that, I make sure to wash potatoes thoroughly before cooking them.  Am I in the wrong here? Should I have properly peeled the potato wedges before baking them? Personally I feel they add more flavor, and from all the pictures on the recipes online, it seems they also do not peel it. Work the Spring There's a book that I read several years ago that I'd like to find.  However, I cannot remember enough about it to find it myself.  Neither the title nor author remain in my mind, nor any keyword or phrase that's unique enough for the Grand Google to nail for me.  Hence my desperate plea here.  In the book, there is an interstellar civilization composed of several power blocks.  The entire civilization did not fill the galaxy, but it did span several stellar systems.  Maybe 10s or 100s.  I seem to remember that one of the civilizations was a decentralized group referred to as the Brotherhood and they seemed to have been keen on preserving old learning and political ideals.  A derelict alien spacecraft from outside all of this had been discovered and at least two of the power blocks sent ships to recover it.  I don't recall if they knew at the time about the power the alien ship had, but they must have had some clue since there was some urgency in capturing this ship.  One of the ships sent to capture the Alien craft was commanded by somebody who was rather shaken up, to say the least.  He had had some big trauma in his past (like... the last ship he captained was annihilated with all hands.) so he was kinda bummed out about things and wasn't really putting his heart into getting this done.  He was assisted by the ship's AI who had a lot of soul-to-soul chats with the captain, and at one point almost relieved him of command because he was just not getting the job done.  I think the ship's name was Boaz.  These were the "good guys".  There was at least one other ship sent from the "bad guys."  The alien craft had some remarkable powers.  The interior volume was apparently infinite and the operators could see and manipulate anything in the universe.  There might have been some example given where all the rats somewhere were found and disposed of.  Whoever had control was tempted by the ole absolute power corrupts absolutely problem, but nobody but nobody could stop them because of their power.  They could eat and multiply within the ship and become real bad-asses.  But alas... (or hurray, depending upon which side you're one) everything fails eventually, and whoever controlled the ship would eventually succumb to some horrible end.  Whereupon the ship became derelict again until some other lucky race found it and repeated the cycle.  The ship had an AI that was real interested in taking the power for itself, but there was some mechanical, spring-loaded, mechanism outside of its control that needed to be worked in order to execute power.  The AI apparently never figured out how to "work the spring" and this made him crazy angry.  The ending is equally foggy for me.  I'm pretty sure the good guys got the ship.  They agonized over what to do with it.  Apparently it was indestructable and nobody wanted to taste the temptation of all that power.  In the end they dumped it on the surface of a neutron star.  There it sat forever more, probably with the crazy AI really pissed off about being there.  If you've read this book, you'll recognize it immediately from these clues.  I'm looking forward to finding out the name of this book.  Thanks! C# equivalent to wildcard imports in Java Is there a way in C# to import everything inside a namespace like there is in Java with the wildcard character?  import java.awt.*; Elliptic Coordinates - Inverting the transformation The standard way to transform elliptic coordinates $(\mu, \nu)$ $\ to$ Cartesian coordinates $(x,y)$:  $x = a \cosh(\mu) \cos(\nu)$  $y = a \sinh(\mu) \sin(\nu)$  Is there any way to get the transformation $(x,y)$ to $(\mu,\nu)$? Meaning is there a way to find:  $\mu = f(x,y)$  $\nu = g(x,y)$  I'm guessing that it would involve $\sinh^{-1}$'s and $\cosh^{-1}$'s, if it was possible to do this at all. I am trying to install Ubuntu 14.04 on FakeRAID along Win7.  Win 7 has been already installed, and dmraid is running.  During the installation I choose to manually partition, this is the structure:  \dev\mapper\isw_gchdghhd_Volume0 \dev\mapper\isw_gchdghhd_Volume0Op1 - NTFS Win7 loader \dev\mapper\isw_gchdghhd_Volume0Op2 - NTFS Win7 partition \dev\mapper\isw_gchdghhd_Volume0Op3 - swap \dev\mapper\isw_gchdghhd_Volume0Op5 - ext4 mounted as \ \dev\mapper\isw_gchdghhd_Volume0Op6 - ext4 mounted as \home   There is another sequence similar to above marked as "linear".  I choose \dev\mapper\isw_gchdghhd_Volume0 as boot device.  When immediately after the installation begins, an error with question marks pops up, and the installation stops. I am currently in the process of learning how to prove statements by induction. For the most part, I understand but I am at most clueless when it comes to prove some inequalities.  For example, I am having a hard time proving that: $n &lt; 2^n$ is true. (Even if it's clear that it is indeed true!)  In the induction step, I am stuck with: $n + 1 &lt; 2^{n + 1}$  I don't know what would be the assumption I should make from this statement.  Anyone could point me in the right direction? Thanks! I have already dual systems : Windows 7 and Ubuntu installed on my laptop. The Windows 7 had been installed when I bought this laptop, and I don't have a CD.  I just installed VitrualBox under Ubuntu, hoping that I could install another Windows in this virtual machine, so that I could run some applications of Windows even under Ubuntu.  But it seems that I need a CD or ISO of Windows to boot inside this virtual machine. Otherwise, there is an error: FATAL: No bootable medium found! System halted. My question is where I could find a CD or ISO?  At the moment, I am downloading ISO from this , but do not know if it will work. By the way, I am not sure if I should pick the 32-bit or 64-bit? As my CPU is 64-bit but my Ubuntu is i386.  Could anyone help? Thank you very much. Can “zealot” have a positive connotation? A zealot is a person who is fanatical and uncompromising in pursuit of their religious, political, or other ideals.  I have never seen this word used with positive connotation, but could it (without causing raised eyebrows)? For example, “Gandhi was a zealot and should be admired by all who believe in non-violence.”  Also, do “zealous” and “zeal” carry the same connotation as “zealot”? In a sentence taken from the Oxford Dictionary:     ‘The inscription on his tombstone in Groombridge Church, where he is buried alongside his three children, bears his original name and no reference to his nom de plume.’   Is ‘buried’ an adjective, a , or a   because it follows the copular verb is?      Predicate nominatives complete only linking verbs. The linking verbs include the following: the helping verbs is: am, are, was, were, be, being, and been; the sense verbs: look, taste, smell, feel, and sound; and verbs like become, seem, appear, grow, continue, stay, and turn.   Or is it a past participle because the clause is in the passive voice?   Does “he is buried” mean currently his body lies in a coffin covered with soil, or  does it mean he has been buried by someone?  If we say ‘he was buried’ it clearly refers to an event that happened and ended sometime in the past. It might even suggest that the burial ground has since been changed. Is it possible to train a neural network without backpropagation? Many neural network books and tutorials spend a lot of time on the backpropagation algorithm, which is essentially a tool to compute the gradient.  Let's assume we are building a model with ~10K parameters / weights. Is it possible to run the optimization using some gradient free optimization algorithms?   I think computing the numerical gradient would be too slow, but how about other methods such as Nelder-Mead, Simulated Annealing or a Genetic Algorithm?  All the algorithms would suffer from local minima, why obsessed with gradient? What's the typical microclimate in the aircraft passenger cabin? I use Microsoft Outlook 2013 for work and Google Calendar for everything else. I'd like to find an easy way to sync my work calendar to my personal calendar. Ideally, this would be a software application that sits in the background and automates the syncing.  Requirements   Syncing should be one-way only: Outlook to Google. Must support repeating calendar items. Must allow me to specify which calendar in Google to sync to. I will set up a separate calendar for work items. No intermediate server. Export is direct.   Nice to Haves   Automated. Syncs in real time or regularly (say, every fifteen minutes or so). 7 segment binary to hex Uses of the definite article (the) in generic noun phrases I was reading a paragraph about lions and I came up with a question about the definite article (the).  Let me tell you first what I know about it.     1->We use the before a singular noun (when we are sure about the noun.    And the listener and the speaker both know about it).      2->We use the before a plural noun (again when we are sure about the noun,   and we are not talking generally).   So, my question is why does the writer use the here as we are not talking about any specific lion or lions. We are talking generally.  The Paragraph:     Living in the grasslands, scrub, and open woodlands of sub-Saharan   Africa, the lion is the second largest cat in the world. It is dwarfed   slightly by the tiger, which is closely related and has a very similar   body type.    Can we re-write the paragraph something like this:     Living in grasslands, scrub, and open woodlands of sub-Saharan Africa,   lions are the second largest cat in the world. They are dwarfed   slightly by tigers, they are closely related and has a very similar   body type.