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Polynomial divisibility is unchanged by coeff. field extension I would like to verify the validity of the following line of thought:  Let $K \subset E$ be a field extension. Let $K[x]$ be the polynomial ring over $x$ and denote $K(x)$ its field of quotients (the rational functions over $K$).  Now assume $f(x)=g(x)h(x)$ with $f(x),g(x) \in K[x]$ and $h(x) \in E[x]$ and all $f$, $g$, $h$ are different than zero. Then $\frac{f(x)}{g(x)} = h(x)$ in $E(x)$. So $h(x) \in K(x)$. But $h(x)$ is a polynomial, so $h(x) \in K[x]$.  Is there any other simpler argument than that?  Added:  Let me be more specific. Let $x^p - a \in K[x]$, with $p$ being prime and $char(K)=p$. Let $b$ be the unique root of $x^p - a$ in its splitting field $K(b)$. Then $x^p - a=(x-b)^p \in K(b)[x].$ Let $m_b(x) \in K[x]$ be the minimal polynomial of $b$ over $K$. Then $m_b(x)$ divides $(x-b)^p$ and so $m_b(x)=(x-b)^l$ and $l$ is the smallest positive integer for which $(x-b)^l \in K[x]$. Let $p = lq+r$ where $0 \ge &lt;l$. Then $x^p-a=(m_b(x))^q (x-b)^r$. Can i conclude that $(x-b)^r \in K[x]$ and so $r=0$ by the minimality of $l$?  Thanks. What exactly happens when you browse a website in your browser? Show that the conditional statement is a tautology without using a truth table I have been attempting to use identities to get to the answer but I am unable to get anywhere. Here is the equation I am trying to prove tautological without using truth tables:  $[(p\rightarrow q) \land (q \rightarrow r)] \rightarrow (p \rightarrow r )$  And here is my attempt at using identities to work my way up. (a link to a page with a table of logical equivalence identities: )  $\lnot [(p \rightarrow q) \land (q \rightarrow r)] \lor (p \rightarrow r)$  $\lnot [(\lnot p \lor q) \land (q \rightarrow r)] \lor (p \rightarrow r)$  $[\lnot (\lnot p \lor q) \land \lnot (q \rightarrow r)] \lor (p \rightarrow r)$  $[(p\land \lnot q) \land (q \land \lnot r)] \lor (p \rightarrow r)$  I don't know where to go from there. Do the *Result objects in Database have a common ancestor? I'm doing some polymorphic programming, and I decided that I wanted to be able to gracefully handle partial saves using SaveResult without blocking the user experience. The code looks roughly like this:  static void logDmlErrors(SObject[] src, SObject[] dest, SaveResult[] results, String message) {     // Transform results into Chatter error messages, send them to admins }   I then went about writing the utility methods that would use this function:  public static void insertRecords(SObject[] src, SObject[] dest, String message) {     logDmlErrors(src, dest, Database.insert(dest, false), message); }   Everything went well, until I came across the delete and undelete methods, which instead use DeleteResult and UndeleteResult. Both of those two classes contain the exact same methods as SaveResult, and yet don't appear to inherit from a common ancestor. This means that I can't just cast to a super-type to get to this information.  Thankfully, Summer 16 apparently introduced support for serializing (and deserializing) these classes as JSON, so I do have a fairly expensive (but accurate) workaround if I want proper polymorphic behavior, but I thought I'd throw it out there and see if someone has an alternative solution to this problem. In the Iron Man movie Tony flies the Iron Man armour from Miami to Afghanistan, roughly 7800 miles.    What is the maximum range of the Iron Man armour?   What sort of speed does the armour cruise at?  How long would the flight from Miami to Afghanistan have taken?   Could Iron Man circumnavigate the globe without stopping, and how long might this journey take him?   EDIT: Just found this (funny) image, New York to Washington in under an hour?!? :) I have an HP ProBook 450 G2. Every time I shut down the computer and turn it back on after a day, the battery level decreases by about 5%. I'm sure that the computer is shut down, not in sleep or hibernate. I removed the battery from laptop and after inserting it, the battery percentage does not change. I tried updating the BIOS, disabling wake on LAN in BIOS options, and calibrating the battery several times.  Is this drain normal? If not, is it a hardware failure or can I fix it with software? What is the simplest way to prove the Earth is round? Assume you've come in contact with a tribe of people cut off from the rest of the world, or you've gone back in time several thousand years, or (more likely) you've got a numbskull cousin.  How would you prove that the Earth is, in fact, round? CreateService get error: ERROR_INVALID_ADDRESS (0x000001e7) I want to update Acrobat Reader for Firefox, but the download has the .bin extension. How can I install it? How can we prove that for every pair $N \in \mathbb{N}$, and natural number $\beta\in [2, \infty)$ there exists a unique set of integers $x_i \in [0, \beta -1]$, $k\in [0,\infty)$ such that:  $$N = \sum_{i=0}^{k}\beta^ix_{i}, \quad $$  It's a question if every positive integer has its unique representation in every base $\beta$. Is the validity of above statement somehow related to Horner's rule? Why is it differential equations exist on an interval instead of a domain? I understand a domain is the set of input elements a function is defined for (and can have breaks in it e.g. union of 2 sets) and a interval is a continuous range of real numbers. Why do we speak of the solution to a differential equation over an interval instead of an domain? If the solution to an ODE is just a function, and we normally speak of functions as having a domain, then why wouldn't we use domain? One of the problems with Bohr's theory to describe the hydrogen atom, was that the electron orbiting around the nucleus has an acceleration. Therefore it radiates and loses energy, until it would collapse with the nucleus.  Now Schrödinger describes the electron as a wave function. His theory is able to describe all atoms (in contrast to Bohr's model), but how is the radiation problem solved? I understand that the wave has not an exact position in the time anymore. But the electron still "moves", so it has an acceleration anyway (because of vibrations or so).   Why is in this theory the electron not radiating anymore? And if it is, why doesn't the atom collapse? On Irix's  4Dwm, there was an ability to minimise windows into a box (contrary to the task bar used by modern window managers). I have seen this also on an old HPUX.  See the "console" square in the linked image:    Is it possible to accomplish on Ubuntu, either with a plugin or maybe some  window manager other than Unity? Add leading zeroes to number in Java? "I saw him crossing" vs. "I saw him cross" I saw him crossing the road.   I saw him cross the road.   Which one is correct and why? Trying to remember a book that had people (I think they turned white?) segregated into a compound Implementing auto correct/complete on Stack Overflow I'm looking for a good way to perform multi-row inserts into an Oracle 9 database.  The following works in MySQL but doesn't seem to be supported in Oracle.  INSERT INTO TMP_DIM_EXCH_RT  (EXCH_WH_KEY,   EXCH_NAT_KEY,   EXCH_DATE, EXCH_RATE,   FROM_CURCY_CD,   TO_CURCY_CD,   EXCH_EFF_DATE,   EXCH_EFF_END_DATE,   EXCH_LAST_UPDATED_DATE)  VALUES     (1, 1, '28-AUG-2008', 109.49, 'USD', 'JPY', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (2, 1, '28-AUG-2008', .54, 'USD', 'GBP', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (3, 1, '28-AUG-2008', 1.05, 'USD', 'CAD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (4, 1, '28-AUG-2008', .68, 'USD', 'EUR', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (5, 1, '28-AUG-2008', 1.16, 'USD', 'AUD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (6, 1, '28-AUG-2008', 7.81, 'USD', 'HKD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'); Find the asymptotic tight bound in  $$ T(n) = 4T\left(\frac{n}{2}\right) + n^{2}\log n. $$  where $ \log n= \log _{2}n $ and $T(1) = 1$.  I should solve this using all three common methods: iteration, master theorem and substitution.  It is highly likely that this kind of recurrence equation will be in my test in two days. Thank you very much in advance! How to write a unit-test / test class for trigger?