This has been asked  but marked as answered and I don’t feel like the question was ever answered, or at least was not clear to me. I don’t understand why the set consisting only of the element $\{0\}$ along with the usual $+$ and $×$ does not satisfy the criteria, since $0$ acts as both the additive and multiplicative identity. That is, letting $G = \{0\}$, then  $∀ g ∈ G, 0+g = g$ and  $∀ g ∈ G, 0·g = g$ (Since $0·0 = 0$ )   Similarly, it is both its own additive and multiplicative inverse. What is the problem at only the field level, without wishing it satisfy some additional properties for category theory or algebraic/arithmetic geometry?
I've heard that in the study of finite fields, and other concepts related to finite fields, mathematicians have found a sort of gap: there are various results and things that seem like they correspond to a field with one element, $\mathbf{F}_1$, even though there is no such field. Of course, there is an object which seems similar to a field with one element: the zero ring, $\mathbf{Z}_1$, defined as having one element, $0$, obeying the equations $$0 + 0 = 0 - 0 = 0 \cdot 0 = 1 = 0.$$ The zero ring seems to behave like a field in every respect, except for the fact that it fails to satisfy $0 \ne 1$. (But the importance of the axiom $0 \ne 1$ isn't clear to me.) However, in any case, there seems to be consensus that the zero ring is not the field with one element. People say that the zero ring &quot;does not behave like a finite field&quot; () or that it &quot;does not have the features that mathematicians need&quot; (). I'm not familiar enough with algebraic geometry to understand the properties that $\mathbf{F}_1$ is expected to have. Is there an elementary, undergraduate-level explanation of  why it seems like a field with one element &quot;ought to exist&quot; in the first place, what properties we expect it to have and why, and how the zero ring fails to satisfy these properties?  Or do I need to study algebraic geometry if I want to have an understanding of any of this? (A side question: do we have a good answer to the question of whether the single element of $\mathbf{F}_1$ ought to be $0$, $1$, both, or neither?)
How to extract a floating number from a string How does new shield system work? Today I saw a sentence:  She surely can do it.  What does that mean? How does it differ from:  She certainly can do it. Surely she can do it.  This slight difference in meaning of those sentences I am not able to notice. Could you please help me? Thanks in advance! Enqueue Stylesheets After Theme's "rtl.css" I am currently developing a theme with internationalization support, and would like to load stylesheets after my theme's rtl.css (an alternative style.css loaded for languages with a right-to-left reading order).  I am familiar with enqueueing my stylesheets and controlling their load order with dependencies using the wp_register_style() and wp_enqueue_style() functions, however rtl.css is loaded automatically by WordPress whenever the locale's language direction is rtl, and there doesn't seem to be a way to specify the stylesheet as a dependency.  I know I can conditionally enqueue stylesheets using the is_rtl() function, however  these stylesheets' &lt;link&gt; elements are still printed out before WordPress automatically prints rtl.css's. I could also simply manually insert &lt;link&gt;s to the stylesheets that I wish to depend on rtl.css right after the call to wp_head() in my theme's header.php file, but I believe it's bad practice to do so, and I would like my theme to be commercial quality.  How can I load stylesheets after WordPress loads rtl.css? Shifting -5 V to +5 V signal to 0 to +5 V for ADC I want to convert the  output voltage of an ECG (-5 to +5) to 0 - +5 volt for ADC. Should i use Op amp ? What is my problem in this picture? I simulated it in Proteus ... but when input voltage is higher than 4 V or lower than -4 V Vout gives an unexpected value.   Below is another simulink that have same problem ...    My idea to make this level of voltage is (Vin +10)/2 that give the range 0-5 V. Take a look at the following picture:   Why does the light at the bottom of the water form this pattern? I have also seen the same phenomenon in all the swimming pools I've ever visited, of whatever depth they may be. How does this happen? Let $f$ and $g$ be entire functions such that $f^n+g^n=1$, where $n\geq 3$ is an integer. Prove that $f$ and $g$ are constant.  I suppose I should somehow prove that either $f$ or $g$ is bounded so that I can apply Liouville's Theorem, but I don't see how. I tried setting the derivative of the left hand side equal to zero and work with that but that did not seem to work. Word for a person who continually tries to teach you when it's not necessary I've experienced this way too many times, and the best example I can give is a person who treats you like you have no idea how to do X and who keeps teaching you how to do it. I feel handicapped. Take for example how to play a card game. You know how to play the game because you've played it 5 times. However, there is a person always insists on explaining the game to you - like you are always new to the game - when you don't actually need any help and you just want to think for yourself. Another example situation is doing a repetitive task which you already have experience with, and yet the person continually tries to 'teach' you when you don't need help/already know how to do it. It almost makes you mad because you really don't need help. As an example which recently happened to me:  &quot;Hey, do you remember how to start the wood chipper?&quot; &quot;Yes I do&quot; &quot;Okay, well, let me show you one more time&quot;  Again, is there a term for someone who does this? Derive a formula to find the number of trailing zeroes in $n!$ Possible Duplicate:          I know that I have to find the number of factors of $5$'s, $25$'s, $125$'s etc. in order to do this. But how can you derive such a formula for any number $n$? How does similar_text work? Can you set custom ringtones for contact groups? In iPhone, you can easily change the default ringtone (Settings → Sounds → Ringtone) and set a custom ringtone for individual people (Contacts → [some person] → Edit → Ringtone).   But can you set a custom tone for a group of contacts? (E.g. "family" or "workmates".)  I think that would be handy—and quite basic functionality, so to speak—but I couldn't find a way to do this through the iPhone UI. Is it possible at all (without jailbreaking)? Let $G$, $H$, and $K$ be abelian groups. Assume $G \oplus H \cong G \oplus K$. If each are finitely generated, then the structure theorem easily gives us that $H \cong K$. However, there are counterexamples in the general case, namely $H=\mathbb{Z}$, $K=\mathbb{Z} \oplus \mathbb{Z}$, and $G=\bigoplus_{i=1}^\infty\mathbb{Z}$.   I am wondering what goes wrong with the following argument:   Form the split short exact sequence  $$0 \to G \xrightarrow{i} G \oplus H \xrightarrow{\pi} H \to 0$$  where $i$ and $\pi$ are the inclusion and projection, respectively. Then don't we immediately get from the first isomorphism theorem that $H \cong (G \oplus H)/G$, which would contradict the counterexample above?   If we run this for the counterexample given, we get the sequence   $$0 \to \bigoplus_{i=1}^\infty\mathbb{Z} \to \bigoplus_{i=1}^\infty\mathbb{Z} \to \mathbb{Z} \to 0$$  which seems to still be short exact, even though here $G \cong G \oplus H$. But of course here we can see directly that $(G \oplus H)/G \cong G/G$, which is trivial. So what went wrong?   It seems to me to be something along these lines: In this case, the kernel of the projection is not the same as the original group, it is just, say, $0 \oplus \mathbb{Z} \oplus \mathbb{Z} \oplus \dots$, i.e. nontrivial from the second coordinate on, which just happens to be isomorphic to the original group. So if we form the other short exact sequence with $K$ in place of $H$, then the difference seems to be that we have two different projection maps with different (but isomorphic) kernels.   But I'm not really sure how to explain this in the general case, if my reasoning is correct to begin with.   Thanks in advance for any help! Error on Ubuntu boot up - "recovering journal" Trying to figure out what this means:    I got into the GRUB menu, chose Advanced Options, boot in recovery mode, login as root, and I did:  root@xps:~# sudo fsck -f / fsck from util-linux 2.27.1 e2fsck 1.42.13 (17-May-2015) /dev/nvme0n1p2 is mounted. e2fsck: Cannot continue, aborting.   Coincidentally, I am also seeing this fuzzy rainbow thing on the outside of desktop windows:    But now the machine won't even boot up at all. Connection pooling in PHP How do I get breading to stick to pre-cooked chicken for pan frying? After 4 years of being a Mac user, there is only one thing that I still miss from the Windows world, and that is being able to switch with a single keystroke between all open windows in the computer.  What is the best app or utility, free or not, for accomplishing that?  I've used , but I never was completely satisfied with it. It's a bit slow and the switcher UI is not very pretty.  Do you know any alternative?    EDIT (2016): This question keeps getting views and upvotes after 6 years. When I asked this question in 2010 I was still transitioning from being a Windows user. Eventually, I got used to Mac OS X UI and nowadays I don't use any third party utility for managing app windows, just Mission Control and the native keyboard shortcuts. So, although this question is no longer relevant for me I hope it is useful to new Mac users making the switch (Including this comment). Prompted by a  I was pondering the idea of bikes with variable gearing (derailleur or otherwise) but without the ability to freewheel. Have such bikes ever been widely used?  They would appear to share many of the advantages and disadvantages of fixies but be better at handling hills (up and down) while being more complex. Prove by induction that $n! > 2^n$ Suppose that when $n=k$  $(k\geq4)$, we have that $k!&gt;2^k$.  Now, we have to prove that $(k+1)!\geq2^{k+1}$ when $n=(k+1)$ $(k\geq4)$.  $$(k+1)! = (k+1)k! &gt; (k+1)2^k  \text{ (since }k!&gt;2^k)$$  That implies $(k+1)!&gt;2^k2$, since $(k+1)&gt;2$ because of $k$ is greater than or equal to 4.  Therefore $(k+1)!&gt;2^{k+1}$  Finally  we may conclude that $n!&gt;2^n$ for all integers $n\geq4$    Solved, thank you! What is a token? Please forgive such a basic question, but I did some Google searching and looked around on this site,  and while there are many threads and online resources that involve tokens, I can't seem to find a direct explanation of what a token is, specifically in the context of TeX, LaTeX, etc. Perhaps there is a good explanation of tokens in the TeXbook, but I don't have a copy of it.   gives me some idea:     A token is a string of characters, categorized according to the rules as a symbol (e.g., IDENTIFIER, NUMBER, COMMA).   but this is a bit too general for me to understand. Taking a guess based on the Wikipedia definition and , is it correct to say that a token is "a thing that has a catcode"? (which, even if true, does not seem entirely satisfactory as an answer: what characterizes the things that are given catcodes?)  Some illustrative examples, and non-examples, of tokens in the realm of TeX, LaTeX, etc. would be much appreciated. Install 18.04 server without network connection