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Hilbert's Irreduicibility Theorem

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Hilbert's Irreduicibility Theorem :  Let $\Q$ be the field of rational numbers and $f_1(x_1, x_2, ..., x_n, y_1, y_2, ..., y_m), f_1(x_1, x_2, ..., x_n, y_1, y_2, ..., y_m),...,f_r(x_1, x_2, ..., x_n, y_1, y_2, ..., y_m)$ be a set of irreducible polynomials in $\Q(x_1, x_2, ..., x_n)[y_1, y_2, ..., y_m]$ . Then Hilbert's Irreduicibility Theorem tells that there exists an n-tuple $(a_1, a_2, ..., a_n)$ in $\Q^n$ such that $f_1(a_1, a_2, ..., a_n, y_1, y_2, ..., y_m), f_2(a_1, a_2, ..., a_n, y_1, y_2, ..., y_m),..., f_r (a_1, a_2, ..., a_n, y_1, y_2, ..., y_m)$ are irreducible in $\Q[ y_1, y_2, ..., y_m]$. A cute application of above theorem is that, If a polynomial $p(x)$ in $\Q[x]$ takes square value at every points in $\Q$ then $p(x)$ is itself a square in $\Q[x]$. (Hint : Apply Hilbert's irreducibilty to f(x, y) = y^2-g(x).)
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Gödel’s Incompleteness Theorems (The Continuum hypothesis)

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  Gödel’s Incompleteness Theorems (Stanford Encyclopedia of Philosophy)
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Centre and Conjugacy Classes of Dihedral Group

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Compactness in Metric Spaces

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  The idea of compactness comes from the fact that these are the nice set next to the finite sets. i.e. It shares many properties of finite sets in a topological space. The first thing common between finite and compact sets is, these are closed and bounded. Every sequence has a limit point. A continuous image of a compact set is compact and many more. Compactness has a very close relation to completeness . Let X be a topological space then the definition of compactness is defined in the following way-: X is said to be compact if every open covering of X has a finite sub-covering. Heine-Borel Theorem -: Heine-Borel make our work easy for Euclidean spaces by generalizing that, ‘ K is a compact subset of R if and only if it is closed and bounded. ’ But above theorem doesn’t work well for all metric spaces , for example -: Take N with discrete metric. What is missing here? Completeness? No! Boundedness? No! Closedness? No! Then what? The thing that is missing is totally bounded
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Riemann’s Rearrangement Theorem

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 In 1827, Peter Lejeune-Dirichlet discovered this surprising result while working on conditions that ensured the convergence of the Fourier series. He was the first to notice that it is possible to rearrange the terms of certain series (now known as conditionally convergent series) so that the sum would change. Why is this result possible?  *ma049.qxd (washington.edu)
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Some Properties of Infinite Sum and Product

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Primality C++ Test Using a Basic Concept of Number Theory

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 Primality testing is some sort of exciting stuff for a beginner in coding. There are various techniques for doing this. Following code is written by me that used the following fact that -: "N is a composite number if it has at least one non-trivial divisor d, such that d is less than or equal to sqrt(N)".  Proof of the fact is simple and can be proved using contradiction(Hint -; N is composite. What if all the non-trivial divisors are greater than sqrt(N)? ). Direct link -:  http://cpp.sh/7sng3 // To detect whether a number is prime or not. // Written by Sushant #include <iostream> #include<cmath> using namespace std; int main() { int p,q,i(0),d(0); cout<<"Enter the number -: \n"; cin>>p; // enter the prime q=int(sqrt(p)); // storing the integer part of square root                   for(i=2; i <= q;i++)                            {                                 // using the fact that if x is a composite number then it must have a non-tr
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