I'm looking to prove the following:  Let $K$ be a field and suppose that $[K(\alpha):K]=p$ and $[K(\beta):K]=q$ for $p,q$ distinct primes. Then $K(\alpha+\beta):K$ has degree $pq$.  Of course I know that $K(\alpha+\beta)$ has either degree $1,p,q$ or $pq$ over $K$. But I do not know how to rule out the cases where the degree is $p$ or $q$. Is there another way?
I am having some trouble with the following problem:     Let $K\subseteq L$ be fields. Suppose $\alpha,\beta\in L$ are algebraic elements over $K$ of degrees $p,q$ respectively, where $p$ and $q$ are distinct primes. Show that $\alpha+\beta$ is algebraic over $K$ with degree $pq$.   I have shown that $\alpha+\beta$ is algebraic over $K$, but I am having trouble showing that the degree is $pq$. I have previously shown that $[K(\alpha,\beta):K]=pq$, so I have been trying to use this result. I thought that if I could show that $K(\alpha,\beta)=K(\alpha+\beta)$ then I'd be done. To show this I first noted that clearly $K(\alpha+\beta)\subseteq K(\alpha,\beta)$. Then we have $$pq=[K(\alpha,\beta):K]=[K(\alpha,\beta):K(\alpha+\beta)][K(\alpha+\beta):K]$$ so that $[K(\alpha+\beta):K]=1,p,q,pq$. I tried looking at the cases when it is equal to $1,p,q$ and deriving a contradiction, but have been unsuccessful.   Another way I thought of solving this is to show that $\alpha,\beta\in K(\alpha+\beta)$, which would allow me to conclude that $K(\alpha+\beta)=K(\alpha,\beta)$, but I am not sure how to complete this either.    I am looking for some assistance to show that $[K(\alpha+\beta):K]\neq 1,p,q$, or that $\alpha,\beta\in K(\alpha,\beta)$. If you have any other solutions to this problem I would like to see those too.
Does Gollum know that others call him Gollum? "You should be well-organised" or "You should be well organised"? I was wondering why a laser beam diverges. If all the photons are in the same direction, I would imagine that it would stay that way over a long distance. I am aware that a perfectly collimated beam with no divergence cannot be created due to diffraction, but I am looking for an explanation based on  photons rather than wave physics. Run a command on all subfolders If you have a series of subfolders (like from a to z) and want to run a command on each one of them (like rm *.pdf or ls *.pdf), how do you do that? The "manual" approach would be cd a, rm *.pdf, or ls *.pdf, cd .., cd b, ... That seems too complicated, so I believe there must be an easier approach. Find html element which id starts with Can "Mr", "Mrs", etc. be used with a first name? I've been reading through a lot of the rookie Java questions on finalize() and find it kind of bewildering that no one has really made it plain that finalize() is an unreliable way to clean up resources.  I saw someone comment that they use it to clean up Connections, which is really scary since the only way to come as close to a guarantee that a Connection is closed is to implement try (catch) finally.    I was not schooled in CS, but I have been programming in Java professionally for close to a decade now and I have never seen anyone implement finalize() in a production system ever.  This still doesn't mean that it doesn't have its uses, or that people I've worked with have been doing it right.  So my question is, what use cases are there for implementing finalize() that cannot be handled more reliably via another process or syntax within the language?    Please provide specific scenarios or your experience, simply repeating a Java text book, or finalize's intended use is not enough, as is not the intent of this question. How to Execute ShortCut From Command line in Windows 7 We came with a scenario where we have to use Shortcut file (.lnk), which is on the desktop, to execute an application for a headless device (i.e, without manual intervention).  Is there any way to execute it from Command prompt? Website Visible only to Registered users Unicode input retrieved via PrimeFaces input components become corrupted "Subject, verb, direct object, object complement" versus "subject, verb, indirect object, direct object" Reading English Grammar (HarperCollins College Outline, published by HarperResource, an imprint of HarperCollinsPublishers) I found a chapter (Sentence Basics) that explains that in English there are different sentence patterns; in particular   Subject, verb, direct object, object complement Subject, verb, indirect object, direct object   The chapter reports the following example for the patterns:     Alice called Charles Professor Miller.   Alice gave Charles the rabbit.   What is the difference between the two kind of sentences? What is the difference between the object complement in the first case, and the indirect object in the second case?  PS: The question, which could appear silly, comes from the fact in my first language there is only the object complement. Limit of derivative is zero. Does it imply a limit for f(x)? When should static_cast, dynamic_cast, const_cast and reinterpret_cast be used? What are the proper uses of:   static_cast dynamic_cast const_cast reinterpret_cast C-style cast (type)value Function-style cast type(value)   How does one decide which to use in which specific cases? If the union of two sets is contained in the intersection, then one is contained in the other ($[A\cup B\subseteq A\cap B]\implies A \subseteq B$) Let $A$ and $B$ be sets. Let $A\cup B\subseteq A\cap B.$ Prove that $A\subseteq  B.$   My understanding of this question is that all the elements of set $A$ that intersects with set $B$ exists in the union of sets $A$ and $B$ and because in order for the elements to intersect, the elements must exist in both sets $A$ and $B$. Therefore set $A$ is a subset of $B$.  My question is how do I write the proof down so that it is an acceptable answer? 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IEEEtran - How to include ORCID in TeX/PDF with PdfLatex This question led to a new package:   I would like to display the ORCID id in an author list in an IEEEtran pdf, like that:  Is there a standard way to include it in an IEEEtran article with PdfLaTeX? I used two approaches from , but I failed. One is to insert the ORCID icon in the author list. I used the blow code: \author{Ali Al-Obaidi \href{https://orcid.org/0000-0000-0000-0000}{\includegraphics[scale=1]{figures/orcid_16x16.png}}}  where orchid_16x16.png is provided in the link . But the result is that the icon is not clear enough, even I replaced it with a different size. The other approach is to use the  package, like that: \usepackage{academicons} \definecolor{orcidlogocol}{HTML}{A6CE39} \author{AAA \href{https://orcid.org/0000-0000-0000-0000}{\textcolor{orcidlogocol}{\aiOrcid}}}  However, the result is like that:  And I also got two errors: &quot;Latex Error: Encoding scheme 'TU' unknown&quot; and &quot;Bad character code (59865)&quot;. In the latest academicons.sty (see ), academicons use the font TU. However, it seems that the compiler PdfLateX cannot support this font. Thus, is there anyone that knows a standard way to include it in an IEEEtran article with PdfLaTeX? Please help me.