I saw it's been answered before but I could not understand the answers as it delved into material that I did not study yet.  My question is as follows:  Suppose $p \in F[x]$ is the minimal polynomial of some matrix $A \in M_n(F)$ and $F \subseteq K$  I need to show that $p$ is also the minimal polynomial of $A$ over the field $K$. In other words: Show that the minimal polynomial remains the same when we move to a bigger field.  What I thought of:  Obviously $p$ is irreducible in $F$, suppose it is reducible in $K$, which means there are $q,t \in K, q,t \notin F$ such that $p=qt$, now we just need to show that $q(A) \neq 0$ and $t(A) \neq 0$...But why is that so?  Edit: Another way of presenting this question is "Show that if $p$ is the minimal polynomial of $A$ over the field $K$ while the entries of$~A$ are contained in a subfield $F \subseteq K$, then all the coefficients of $p$ are elements of $F$".
I think I have heard that the following is true before, but I don't know how to prove it:     Let $A$ be a matrix with real entries. Then the minimal polynomial of $A$ over $\mathbb{C}$ is the same as the minimal polynomial of $A$ over $\mathbb{R}$.   Is this true? Would anyone be willing to provide a proof?    Attempt at a proof:  Let $M(t)$ be the minimal polynomial over the reals, and $P(t)$ over the complex numbers. We can look at $M$ as a polynomial over $\Bbb C$, in which case it will fulfil $M(A)=0$, and therefore $P(t)$ divides it. In addition, we can look at $P(t)$ as the sum of two polynomials: $R(t)+iK(t)$. Plugging $A$ we get that $R(A)+iK(A)=P(A)=0$, but this forces both $R(A)=0$ and $K(A)=0$. Looking at both $K$ and $R$ as real polynomials, we get that $M(t)$ divides them both, and therefore divides $R+iK=P$.   Now $M$ and $P$ are monic polynomials, and they divide each other, therefore $M=P$.  Does this look to be correct?    More generally, one might prove the following     Let $A$ be any square matrix with entries in a field$~K$, and let $F$ be an extension field of$~K$. Then the minimal polynomial of$~A$ over$~F$ is the same as the minimal polynomial of $A$ over$~K$.
I'm just wondering how it is possible that IGN.com has this in its title tag:      Video Games, Cheats, Walkthroughs, Game Trailers, Reviews, News, Previews &amp; Videos at IGN   while on Google its title is simply "IGN".   How can one achieve such a thing? Does it use some special meta tag I can't see? Can I remove receipts from imported products from my passport? I have recently imported books into Canada from Japan.  As I bought in excess of 5000¥, the purchase was tax free and the receipt was stapled to my passport and stamped.  Since I've now come back to Canada and no issues occurred with my imports, Is it safe to remove the receipts from my passport?  As it was stamped and stapled, there will remain a trace that something was there. Could this cause problems when I return to Japan at a later time? How can timelapse sequences be shot without in-camera support? How can timelapse sequences be shot without having in-camera support for it?  Which third-party tools are recommended? $E$ is a measurable set of finite measure and $1 \le p_{1} &lt; p_{2} &lt; \infty$. Consider the linear space $L^{p_{2}}(E)$ normed by $||.||_{p_{1}}$. Is this space Banach? What software is focused on reviewing and organizing images? Something small goes wrong, and this triggers something slightly bigger, which triggers something slightly bigger, and so on and so forth until you end up with a chain of problems of increasing severity until it ends in catastrophy.  Something like the old saying "for want of a nail the battle was lost", only highlighting that each intermediate issue was a problem in its own right, not just a stepping-stone to the final disaster.  Is there a common word/phrase that specifically describes this? I'm thinking "cascade of ever-increasing problems", but this seems clumsy, and I suspect there might already be a neat word or expression that covers it. How to show that $C=C[0,1]$ is a Banach space Let $C=C[0,1]$ be the space of all continuous functions on $[0,1]$. Define $\|f \|=\max \ |f(x)|$. I want to show that $C$ is a Banach space.     Below is my attempt and I was wondering if it's ok.    I know I have to show that $C$ is a complete normed space. Clearly, $\|f\| \geqslant 0$ and $\|f\|=0 \Leftrightarrow f=0$. $\|cf \|=\max~|cf(x)|=|c|\max |f(x)|=|c| \cdot \|f\|$. $\|f+g\|=\max~|f(x)+g(x)|\leq \max~|f(x)|+\max~|g(x)|=\|f\|+ \|g\|$. So $C$ is a normed space.   Next, I show that every Cauchy sequence in $C$ is convergent. Let $\{f_n\}$ be a Cauchy sequence in $C$. Let $\varepsilon \gt 0.$ Then $\exists$ an $N_1$ such that $$ \max~|f_n(x)-f_m(x)| \lt \frac{\varepsilon}{2}$$ for $n, m \gt N_1$ and $x\in[0,1]$.  But there is a subsequence $f_{k_n} $, which converges to $f$. So $\exists$ an $N_2$ such that $$ \max~\left|f_{k_n} - f\right|\lt \frac{\varepsilon}{2}$$ for each $n\gt N_2$.  Now Let $N = \max\{N_1, N_2\}$, if $n \gt N$ then $k_n \geqslant n\gt N$. So we have $$ \max~\left|f_n(x) - f(x)\right| \leqslant \max~\left|f_n - f_{k_n}\right| + \max~\left| f_{k_n} - f\right| \lt\frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon.$$  Thus, $\|f_n-f\| \to 0$ as $n\to \infty$. $\quad \square$  Thanks. What is the equation for the scale factor of the universe, a(t), for the best fit of data to the $\Lambda CDM$ Model of Cosmology? Ideally I like a single formula or multiple formulas for different time ranges that would cover the time from the end of inflation through 100+ billion years after the big bang using  the $\Lambda CDM$ Model.  I know that from the end of inflation back to the time of the big bang would be much more speculative, but some wild estimate would be appreciated for that time range also!  The Friedmann–Lemaître–Robertson–Walker metric defines the scale factor, a(t), from the metric: $$-c^2d\tau^2 = -c^2dt^2+a(t)^2d\Sigma^2$$ where $d\Sigma^2$ ranges over the 3 dimensional space of the universe and does not depend on time. Usually the scale of the scale factor is set by defining  $a(t_{now}) = a(13.78 B yr) \equiv 1$ The locus of the intersection point of two perpendicular tangents to a given ellipse Changing internal URL path for SEO I have a site where Google has already indexed about 1 million pages, and I want to change all the URL paths for SEO purposes, ensuring that we always have our keywords in the path.  Technically I know how to complete it (301 redirects, new canonical URL, update sitemap, etc.) but I'm wondering if Google will penalize the site because of it, or if links will disappear from search results until they are re-crawled? Can Windows use a Linux shell instead of cmd? Proving a Lipschitz function is continuous A function $f:D\subset \mathbb R \to \mathbb R$  is lipschitz given that there exists a $L\gt0$ such that $|f(x)-f(y)|\le L|x-y|$  I need to prove this function is then continuous. Is there a best definition of continuous functions to use for this proof given the definition of Lipschitz?  Is the epsilon delta way the only way to do it? can someone help me set it up? I really struggle with implementing that definition. When should I use \input vs. \include? Where can I find computer modern fonts for use in other programs, e.g. Adobe Illustrator? The ideal $I= \langle x,y \rangle\subset k[x,y]$ is not a principal ideal.   I don't know how to consider it. Any suggestions? Global keyboard capture in C# application I want to capture a keyboard shortcut in my application and trigger a dialog to appear if the user presses a keyboard combo even outside of the app. Similar to Google Desktop Search's Ctrl, Ctrl to bring up the search dialog.  I have tried using some keyboard hook modules out there that basically use Win32 interop to get this effect but each implementation I've tried ties down the keyboard to some extent to where you start getting weird behaviors when the application is doing something intensive. Such as loading a large amount of data, this would cause the keyboard and mouse to lockup.  I'm looking for a lightweight solution that would allow this to be done without tying down the keyboard and mouse. What are suitable TV sizes and viewing distances for couch surfing on a 1080p LCD HDTV? How can I prove the formula for calculating successive entries in a given row of Pascal's triangle? I've found in Wikipedia the formula for calculating an individual row in Pascal's Triangle: $$v_c = v_{c-1}\left(\frac{r-c}{c}\right).$$ where $r = \mathrm{row}+1$, $c$ is the column starting from $0$ on left and $v_0 = 1$.  Now, I've tried to do by hand and it works,  but I don't understand how to find this magic formula when I don't have access to Wikipedia...:) Although many have attempted to explain the difference among \mathrm, \textrm, \textnormal and \text (and others?), I found them misleading.  See for example the LaTeX's Wiki \mathrm explanation   Why is it wrong? Let's try all of them within an article and a beamer and you'll see the difference and, therefore, error.  equation.tex  \begin{eqnarray*}   \int_1^9\! \cos{x} \,\mathrm{d}x &amp; &amp; \textrm{this is textrm}\   \sum_1^9 y                       &amp; &amp; \textsf{this is textsf}\   \prod_1^9 z                      &amp; &amp; \textnormal{this is textnormal}\   \bigcup_1^9 w                    &amp; &amp; \text{and this is just text} \end{eqnarray*}   article.tex  \documentclass{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb}  \begin{document}   \input {equation} \end{document}     And until here everything looks fine...  beamer.tex  \documentclass{beamer} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb}  \begin{document}   \begin{frame}     \input{equation}   \end{frame} \end{document}     And here we can clearly see that \mathrm does not behave as expected, i.e. does not provide a function aspect to the d. Does anyone know what am I supposed to use here instead? Just \text?? o.O  Moreover, I cannot see any difference between \textnormal and \text.  Finally, \textrm looks useless.. Finding the limit of $\left(\frac{n}{n+1}\right)^n$ Find the limit of: $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$   I'm pretty sure it goes to zero since $(n+1)^n &gt; n^n$ but when I input large numbers it goes to $0.36$.  Also, when factoring: $$n^{1/n}\left(\frac{1}{1+\frac1n}\right)^n$$ it looks like it goes to $1$.   So how can I find this limit?