Can we define any kind of addition and multiplication on the set of all non-negative integers such that it becomes a field. I think not. Can we prove this ? If we have only a finite collection with prime cardinality then only it may become a field.
To make sure that we are talking about the same, I would like to post the relevant definitions I know first.  Definitions:  A pair $(G, +)$ where $G$ is a set and  $+: G \times G \rightarrow G$  is called a commutative group if it has all of the following three characteristics:   (associativity): $\forall x,y,z \in G: (x+y)+z=x+(y+z)$ (identity element): $\exists e \in G \forall x \in G: e + x = x = x +e$ (inverse elements): $\forall x \in G \exists x^{-1} \in G: x^{-1} + x = e = x + x^{-1}$ (commutativity): $\forall x,y \in G: x+y=y+x$.     A triple $(\mathbb{K}, +, \cdot)$, where $\mathbb{K}$ is a set and  $+: \mathbb{K} \times \mathbb{K} \rightarrow \mathbb{K}$  $\cdot: \mathbb{K} \times \mathbb{K} \rightarrow \mathbb{K}$  is called a field if it has all of the following three characteristics:   $(\mathbb{K}, +)$ is a commutative group $(\mathbb{K} \setminus \{0\}, \cdot)$ is a commutative group distributive properties: $\forall x,y,z \in \mathbb{K}: x \cdot (y+z) = (x \cdot y) + (x \cdot z)$ and $\forall x,y,z \in \mathbb{K}: (y+z) \cdot x = (y \cdot x) + (z \cdot x)$     $\mathbb{N^+} := 1, 2, 3, 4, 5 ...$ (all positive integers)  $\mathbb{N}_0 := \{0\} \cup \mathbb{N}$ (the natural numbers with zero)  $\mathbb{N}$ can be both: $\mathbb{N^+}$ or $\mathbb{N}_0$.    My question: Does a field exist that has $\mathbb{N}$ as its set?  I know that $(\mathbb{Q}, +, \cdot)$ is a field and I know that $(\mathbb{N}, + )$ is only a commutative semigroup.   But maybe it's possible to define two mappings $\circ, *$ for $\mathbb{N}$ in such a way that $(\mathbb{N}, \circ, *)$ is a field.    Related:     The smallest field containing the integers is the field of rational   numbers.   Source: en-Wikipedia:   I don't know if this is true. If somebody can prove (or at least scribble the proof of) the quote it would be a good answer, I guess.
Portrait for phone, landscape for Tablet (Android-Layout) How to perform a logistic regression for more than 2 response classes in R I want to predict the impact of oil price over a Colombian oil company's stock price. I plan to use a logistic regression for this with a categorical variable (Up or Down given the direction of the stock price). Here is part of my dataset:  Minute  ecopet  profit  sum_profit   direccion  cl1_chg   sum_cl1    direccion_cl1 571     2160     0       10           Up         -0.03     0.00      Down 572     2160     0        0           Neutral     0.07    -0.03      Down 573     2160     0       -5           Down       -0.08     0.04      Up 574     2160     0       -5           Down       -0.07    -0.04      Down 575     2160     5       -5           Down       -0.08    -0.11      Down 576     2165     0       -25          Down        0.00    -0.19      Down 577     2165     0       -25          Down       -0.05    -0.19      Down 578     2165     0       -15          Down       -0.17    -0.24      Down 579     2165     5       -15          Down       -0.06    -0.41      Down 580     2170     0       -20          Down        0.03    -0.47      Down 581     2170    -10       0           Neutral     0.04    -0.44      Down   My dependent variable is 'direccion'. But as you can see it has 3 response classes. I know that to implement a binary logistic regression in R the code is:  glm.fit=glm(direccion~direccion_cl1, data=datos, family=binomial)   I am working with intraday information and plan to predict what happens when the oil moves up/ down (in the previous 10 minutes) and how it impacts the stock price in the next 10 minutes.  Could anyone tell me how could I perform this? I don't really know how to perform the logistic regression with 3 response classes. Thanks! Is there any other way to do a clean install withot formatting /home? I think my Ubuntu is a bit broken from messing around with it.  Unfortunately during installation, I did not make a separate partition for /home. Now I need to re-install but want to keep my music and films.  Any ideas... Origin and correctness of “ain’t no”? In contemporary American English usage, I come across sentences like:   I ain’t got no money.  Ain’t no man like him.    Saying ain’t no sounds incorrect to me because it is a double negative.  What is the origin of this particular construction? I wasn’t able to find anything online related to the origins of this incorrect usage. Why on earth are o-scopes earth referenced? Thinking about it: You would never find a "Grounded" multimeter as robust and useful if a path to ground through the multimeter were introduced, modifying the circuit's behaviour and possibly damaging the multimeter with currents.  Why are so many oscilloscopes earth referenced? Upon reading some educational material, a majority of the "common mistakes made by students" are placing the grounding clip incorrectly and causing poor results - when the o-scope is just being used as a fancy voltmeter!  I've heard of a Tek scope having an isolation transformer within.. however ignoring that, and taking in to account that newer DSOs may have plastic cases (isolated from you most importantly I would assume) could I just remove the earthing pin, and install a 1:1 AC transformer inbetween the o-scope and outlet and be on my merry way probing various hot/neutral/earthed sources with no worries about a path to ground any longer through it? I have been playing for almost 2 days, and today it is asking me to sign up again.  I have all my game saved in my google account, but I wonder if I just log in from a trainer account will I keep my progress? Ubuntu copy dialogue stops at the last moment for a long time Whenever I try and copy a file to an USB flash disk, the progress bar runs till the end very fast and stops for an indeterminable time before it is complete.  Is there an issue with ubuntu or my flash disk is what I am not able to figure out. All I want is a realistic copy dialogue An inequality involving 4 real or complex numbers What is the Python equivalent of static variables inside a function? I use Stack Overflow a lot and have noticed a certain trend that I myself got caught up in at one time of using the phrase "I am having a problem" in place of "I have a problem."  I would use this phrase for an event in the future, for example, "I am having a steak dinner tonight," but for a problem that I need solved and am asking for a solution, I'd use "I have a problem..."   What would force us to use this phrase? Is it even valid English for the present tense? Installed Ubuntu alongside Windows 8, and W8 won't boot anymore Here's an example of a short conversation between me and a native speaker via text messages:  A: I'm so exited for this app B: Yeah, me too A: We start tomorrow B: OK   Shouldn't it be "We are starting tomorrow"?! Why Multiplication of zero by any number is zero? If I multiply $a$ by $0$, it means $a$ is added $0$ times, but I am not able to visualize $0$ times! My Young's slits experiment doesn’t work. I mean I tried to see wave effect of light but I didn’t see any thing expect particles! I’ve sent you a pic of my laser and a short video of the experiment. Is there any problem you think? I have to find an easy ring which this ring is isomorphic with.  I can prove that $5 = (2+i)(2-i) \in (2+i)$ but this doesn't help me further.  My book says the answer is $\mathbb {Z_5} $ My months and days in my File Manager are in German, how do I change that? On the photo you see 'Dez' : It means Dezember (German for December). I also saw one day Mittwoch (German for Wednesday). How do I change that German issue? This is only my File Manager and not completly but partially. Weird. Find Max Value for each month for the last 3 months, properly Better Information On Defense Damage Clash of Clans The way that Clash of Clans chooses to convey information about how much damage defense buildings output is bad. The current method of telling the user that a mortar does X dmg a second is pretty worthless since they fire much slower than once a second.  Where can I find a break down / what is the damage break down for how much damage these buildings do per shot rather than per second?  Sure, information on each level of the buildings would be nice but if that is too much I think simply providing the data for level one of each building would also work. Execute shell script or command on idle time I'm searching for the possibility to execute a shell script or command based on idle time in the bash shell.  As an example, if I'm in a SSH session to a server of mine and after X minutes of inactivity in the terminal, I want a command to start a desired program like cmatrix, and when I come back and press a key I will get back where I left. Will metering from a DSLR apply to a film SLR? I still like to shoot with film (I use a Nikon F3) but I like the immediate feedback of digital.  I'm thinking of buying a used DSLR to help me get the framing and lighting right when I shoot static subjects.  Then I'd take the final shot with my F3. But would the metering be transferable?  If the shot looks perfect at 1/125 with f/2 on a DSLR, will the same settings work on a film camera?