Show that an algebraic closed field must have infinite many elements.  Let's suppose that an algebraic closed field $K$ has finite many elements.  But how could I get a contradiction??
Show that an algebraically closed field must be infinite.    Answer  If F is a finite field with elements $a_1, ... , a_n$ the polynomial $f(X)=1 + \prod_{i=1}^n (X - a_i)$ has no root in F, so F cannot be algebraically closed.  My Question  Could we not use the same argument if F was countably infinite? Couldn't we say that if F was a field with elements $a_1, a_2, ... $ then the polynomial $f(X) = 1 + \prod_{i=1}^{\infty} (X - a_i)$ does not split over F?  Thank you in advance
Solving recurrence relation. Recurrence In Java, you can qualify local variables and method parameters with the final keyword.  public static void foo(final int x) {   final String qwerty = "bar";  }   Doing so results in not being able to reassign x and qwerty in the body of the method.  This practice nudges your code in the direction of immutability which is generally considered a plus. But, it also tends to clutter up code with "final" showing up everywhere. What is your opinion of the final keyword for local variables and method parameters in Java? Error sending edittext in java server. See code Guest (not logged in) users can only sort questions with specific tag by newest I think this is platform wide (all SE sites).   Logout Choose a tag, eg.  on SO Try to sort questions by something other than newest   No matter what sorting is chosen, questions are sorted by newest How do you get a bad transcript past Ph.D. admissions? I have a master's degree in International Studies, and a double major with Computer Science from undergrad.  My transcripts suck.  There's no other way to dress it up.  I have pretty good teaching experience, and my GRE scores are awesome, and I suspect my recommendations are as bland as everyone else's.  Basically, to an admissions committee, I suspect I'm the model of a student who is probably smart enough but didn't work hard enough.  I want to do a Ph.D. in Political Science, but the response from my applications is looking pretty grim.  Am I permanently out of the running, or is there something I can do for the next few years which will help to counterbalance my unfavorable GPA? Specify a bijection from [0,1] to (0,1]. I upgraded my Nexus 4 to Lollipop a few days ago but I don't like it. I'll leave out my reasons because they're not relevant here. I wiped my phone and installed a clean stock 4.2.2 version instead. I'm happy with this.  Not long after, Android automatically downloaded the 4.3 upgrade and notifies me with the choices "Later" and "Install". I don't want to do this upgrade.  How do I:   get rid of the notification, and  remove the downloaded update (to free up precious storage space).     I have already tried choosing "Later" but the same notification immediately reappears.    Based on Google search results, I also tried going to Settings &gt; Apps &gt; All &gt; Google Services Framework and unticked "show notifications" but it still appears. Force stop or disable also doesn't seem to help, so I've re-enabled the service including notifications. I didn't see any other apps in this list that sounded relevant - did I just not see it? Determining the default Method used in optimization and root-finding algorithms In how many different ways can you split $10$ people into two groups of $5$ people? Is there a built-in checksum utility on Windows 7? Is there a built-in checksum/hash utility on Windows 7? As I understand it, there are several models for what the universe could look like from the “outside”. I don’t know if any of these are outdated, so if they are please tell me if they are.   Expanding bubble. This model says the universe will never stop expanding, and the universe will die because the energy all turns to photons and gets spread out so much. Bubble that will eventually collapse. In this model the universe will stop expanding, and collapse. Everything merges into super-massive black holes, and the universe dies with fire. The universe is a torus. I'm not clear whether the size of the universe in this model is changing, but the central idea is that space is donut-shaped.   Those are the major ones that I have heard of. My question involves what I have heard to be the most popular of the three, #2. In this model, eventually something will "touch" the edge of the universe. Obviously, the first thing to "touch" it would be photons as they travel at, well, the speed of light. This involves all forms of energy in the universe, but I thought it might be easier to explain for something without mass.  What happens if some form of energy "touches" the edge of the universe? Logically, the energy cannot be lost without violating the law of conservation of energy. The only thing I can think of, is that it would "bounce" or be refracted off the edge.  I realize that it sounds silly, however if model #2 is correct then this situation will arise some number of billions of years from now.  I can only assume that there is either a new model that accounts for this, there is some complex explanation, or that model #2 is incorrect. Show that if $f(z)=\sum\limits_{n\geq0}a_nz^n$ is analytic in $\{z\in\mathbb{C}:|z|&lt;1\}\cup\{1\}$ and $\forall n\geq0:a_n\geq0$ then the radius of convergence of the power series is strictly larger than $1$.  My approach:   Let $D=\{z\in\mathbb{C}:|z|\leq1\}$.  We have that $f(1)$ converges, so $\sum\limits_{n\geq 0} a_n&lt;\infty$. Moreover since the $a_n$'s are all positive we have that $|a_n|=a_n$, therefore $\sum\limits_{n\geq 0} |a_n|=\sum\limits_{n\geq 0} a_n&lt;\infty$.  Let $f_n(z)=a_nz^n$, then $\forall z\in D, n\geq 0$ we have:  $$|f_n(z)|=|a_nz^n|=|a_n|\cdot|z^n|=a_n\cdot |z|^n\leq a_n\cdot 1^n\leq a_n$$  Then, applying Weierstrass M-test we get that the power series is convergent in $D$.  Now, I need to "find" an open ball $B$ such that $D\subset B$ and the power series is convergent in it.  Edit  $f$ is analytic in $1$, so there is an open set $U$ that contains $1$ such that   $$f(z)=\sum\limits_{n\geq 0} b_n(z-1)^n$$  Does it mean that the series is convergent in $U$?   I can see that we have that the two series coincide in $D\cap U$. Add/Delete table rows dynamically using JavaScript Community user "stealing" tags? finding and replacing elements in a list Given fields $K\subseteq L$, why does $f,g$ relatively prime in $K[x]$ imply relatively prime in $L[x]$? $K,L$ are fields, $K\subseteq L$. $f,g \in K[x]$. Suppose that $f,g$ are relatively prime as elements of $K[x]$. Prove they remain relatively prime  in $L[x]$.  I've tried everything I can think of. I feel like working with the contrapositive may be helpful but that's just a feeling. Transparent textured planes produce black artifacts Does equality of characteristic polynomials guarantee equivalence of matrices? I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and then the question.  True or False:   let $A,B \in \mathbb{Q}^{n \times n }$  Suppose $xI-A$ and $xI-B$ are equivalent then $\det(xI-A) = \det(xI-B)$.     I wrote in my notes this is true because any two matrices are equivalent in $\mathbb{Q[x]}^{n \times n}$ if and only if they have the same invariant factors.  Since the characteristic polynomial is a product of the invariant factors it follows that $\det(xI-A) = \det(xI-B)$.        True or False:   let $A,B \in \mathbb{Q}^{n \times n }$. If $\det(xI-A) = \det(xI-B)$ (in $\mathbb{Q}[x]$) then $xI-A$ and $xI-B$ are equivalent  in $\mathbb{Q}[x]^{n \times n}$   I think this is false because of the fact that equality of determinants is not strong enough to guarantee all invariant factors are equal but I do not have an example. We know Euler formula:  $$ e^{ix}=\cos(x)+i\sin(x). $$  Does this formula work if we replace real number $x$ with complex number $z$? Can someone give a simple explanation as to why the       $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$   doesn't converge, on the other hand it grows very slowly?   I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks.