User pritam - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:10:54Z http://mathoverflow.net/feeds/user/23980 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/131057/doubt-in-the-proof-of-stickelbergers-theorem Doubt in the proof of Stickelberger's Theorem pritam 2013-05-18T15:35:21Z 2013-05-18T22:54:13Z <p>I was going through the proof of Stickelberger's Theorem, as given in the book 'Algebraic Number Theory' by Richard A Mollin, and I am having some problem in understanding the proof. I will state the theorem and the proof and I will be highly grateful if anyone can explain my doubts. I have also asked this question in MSE( <a href="http://math.stackexchange.com/questions/394785/proof-of-stickelbergers-theorem" rel="nofollow">http://math.stackexchange.com/questions/394785/proof-of-stickelbergers-theorem</a>) but have not got any answers.</p> <p>$\textbf{Theorem :}$ If $K$ is an algebraic number field then $\Delta_K$, the discriminant of $K$, satisfies $$\Delta_K\equiv 0,1\pmod{4}$$</p> <p>$\textbf{Proof :}$ Let $\lbrace a_1,\ldots ,a_n\rbrace\subseteq\mathfrak{O}_K$ be an integral basis for $K$ and $\sigma_1,\ldots\sigma_n :K\to \mathbb{C}$ be all the embeddings of $K$. Then we have by definition, $$\sqrt {\Delta_K}=\det([\sigma_i(a_j)])$$ and this can be written as $$\sqrt{\Delta_K}=\sum_{\pi\in A_n}\prod_{i=1}^n\sigma_i\left(a_{\pi (i)}\right)-\sum_{\pi\not\in A_n}\prod_{i=1}^n\sigma_i\left(a_{\pi (i)}\right):=P-N$$ Now for each embedding $\sigma_i$ we have, $$\sigma_i(P+N)=P+N,\hspace{5mm} \sigma_i(PN)=PN$$ and hence $P+N, PN\in\mathbb{Q}$.</p> <p>Hence we have $P+N,PN\in\mathbb{Z}$, because $P$ and $N$ are both algebraic integers. Now using the identity $$(P-N)^2=(P+N)^2-4PN$$ it follows that $\Delta_K\equiv0,1\pmod{4}.$</p> <p>$\underline{\textbf{My questions}}:$ </p> <p>$(1)$ How can we apply $\sigma_i$ to $P+N$ and $PN$, I mean how does it follow that $P+N, PN\in K$ ?</p> <p>$(2)$ Why is $\sigma_i(P+N)=P+N$ and $\sigma_i(PN)=PN$ ?</p> <p>$(3)$ From the above how does it follow that $P+N, PN\in\mathbb{Q}$ ?</p> http://mathoverflow.net/questions/120733/which-surfaces-can-be-completely-defined-by-a-single-parameterization Which surfaces can be completely defined by a single parameterization? pritam 2013-02-04T06:46:05Z 2013-02-04T07:09:49Z <p>It can be easily shown that any closed and bounded surface of $\mathbb{R}^3$ cannot be covered by a single surface patch, i.e. cannot be homeomorphic to an open set of $\mathbb{R}^2$. What can be said about the non-compact surfaces? Is it possible to characterize all the surfaces of $\mathbb{R}^3$ which are homeomorphic to an open set of $\mathbb{R}^2$?</p> http://mathoverflow.net/questions/119448/self-complementary-cartesian-products Self complementary cartesian products pritam 2013-01-21T07:03:08Z 2013-01-21T07:03:08Z <p>Given two graphs $G$ and $H$ is there a nice way to check whether the cartesian product $G\Box H$ is self complementary without directly computing its complement and searching for isomorphism? For example, how can one show that $K_3\Box K_3$ is self complementary?</p> http://mathoverflow.net/questions/108726/getting-a-bound-on-the-coefficients-of-the-factor-polynomial Getting a bound on the coefficients of the factor polynomial pritam 2012-10-03T17:36:21Z 2012-10-03T23:27:58Z <p>Suppose $f(x):=a_0+a_1x+\cdots+a_nx^n$ is a polynomial in $\mathbb{Z}[x]$ and $|a_i|\leq M$ for each $i=0,\ldots ,n.$ Now suppose $g(x)$ is a factor of $f(x)$ in $\mathbb{Z}[x]$, then is it possible to get a bound on the coefficients of $g(x)$ in terms of $M$ i.e. if $g(x)=\sum_{i=0}^mb_ix^i$ then does there exist some $M^\prime$, which depends only on $M,n$ and $m$, such that $|b_i|\leq M^\prime$ for all $i=0,\ldots ,m$ ?</p> http://mathoverflow.net/questions/106480/which-polynomials-are-solvable-by-radicals Which polynomials are solvable by radicals ? pritam 2012-09-06T06:16:22Z 2012-09-06T06:57:46Z <p>Suppose $n\geq 5$ and $f(x)$ is a polynomial of degree $n$, then in general $f(x)$ is not solvable by radicals, but there are certain special polynomials which are solvable by radicals. Is there a characterization of all such polynomials ?</p> <p>Edit: Of course Galois group is solvable is one characterization, but I am looking for more stronger charaterization for some special values of $n$ (say $n=5$, or $n$ is prime), if not for general $n$.</p> http://mathoverflow.net/questions/105907/why-is-the-physical-space-equivalent-to-mathbbr3 Why is the physical space equivalent to $\mathbb{R}^3$ pritam 2012-08-30T07:07:34Z 2012-08-30T08:38:35Z <p>I am trying to understand what would be the logical reasons behind our assumption that our physical space is equivalent to $\mathbb{R}^3$ or 'physical straight line' is equivalent to $\mathbb{R}$. </p> <p>$\mathbb{R}$ is a basically an algebraically constructed set, which is nothing but the completion of $\mathbb{Q}$, the set of rationals. Now my question is what is the reason behind our approximation of the physical space by this abstract set. Why is this approximation assumed to be most suitable or good approximation ? </p> http://mathoverflow.net/questions/105543/algorithms-to-find-irreducible-polynomials-of-a-given-degree Algorithms to find irreducible polynomials of a given degree pritam 2012-08-26T15:37:11Z 2012-08-26T17:42:30Z <p>I need to know what are the efficient algorithms to find all the irreducible polynomials of a given degree, say $d$ over a given finite field, say $\mathbb{F}_{p^n}.$</p> <p>One way is to factorize the polynomial $x^{p^{dn}}-x$, which is the product of all irreducible polynomials whose degree divides $d$, using factorization algorithms and collect all the degree $d$ factors. But I guess we are doing some extra job here. Are there better algorithms to find all irreducible polynomials of degree $d$ ?</p> <p>I also want to know about the algorithms to find one irreducible polynomial of a given degree over a given finite field.</p> http://mathoverflow.net/questions/105299/diametrically-opposite-points-go-to-diametrically-opposite-points-under-stereogra Diametrically opposite points go to diametrically opposite points under stereographic projection pritam 2012-08-23T06:06:33Z 2012-08-23T09:32:00Z <p>I asked this question in MSE here: <a href="http://math.stackexchange.com/questions/184524/diametrically-opposite-points-go-to-diametrically-opposite-points-under-stereogr" rel="nofollow">http://math.stackexchange.com/questions/184524/diametrically-opposite-points-go-to-diametrically-opposite-points-under-stereogr</a> but I didnâ€™t get an answer. I really need to solve this problem and believe me this is not a homework.</p> <p>Suppose $P_1$ and $P_2$ are two diametrically opposite points of a circle $C$ in the complex plane and suppose $\sigma (z)$ denotes the image of $z\in\mathbb{C}$ on Riemann sphere due to the inverse of the stereographic projection. Then I need to prove that $\sigma (P_1)$ and $\sigma (P_2)$ are also diametrically opposite points of the circle $\sigma (C)$. </p> <p>Intuitively this is obvious but I need to prove this fact algebraically (i.e. using coordinate geometry ). </p> http://mathoverflow.net/questions/98565/primary-ideals-of-the-polynomial-ring Primary ideals of the polynomial ring pritam 2012-06-01T11:35:32Z 2012-06-01T14:09:00Z <p>Is it possible to classify all the primary ideals of the polynomial ring $K[X_1,\ldots ,X_n]$ where $K$ is a field.</p> <p>Or, give a big class of examples of primary ideals which are not prime ideals.</p> http://mathoverflow.net/questions/131057/doubt-in-the-proof-of-stickelbergers-theorem/131068#131068 Comment by pritam pritam 2013-05-18T18:11:52Z 2013-05-18T18:11:52Z Can you please tell how do you prove your first and second claim ? http://mathoverflow.net/questions/120733/which-surfaces-can-be-completely-defined-by-a-single-parameterization Comment by pritam pritam 2013-02-04T07:13:13Z 2013-02-04T07:13:13Z May be 'characterize' is better, edited. And I guess the use of 'cover' is not much relevant in the view of my final question. http://mathoverflow.net/questions/119448/self-complementary-cartesian-products Comment by pritam pritam 2013-01-22T06:00:33Z 2013-01-22T06:00:33Z @Chris godsil: Yes, thats why I said without computing the complement, may be using some arguments on the degrees of verices and using the fact that it is a cartesian product; and this is not a homework. http://mathoverflow.net/questions/106480/which-polynomials-are-solvable-by-radicals Comment by pritam pritam 2012-09-06T07:45:12Z 2012-09-06T07:45:12Z Yes, an easily computable condition on polynomials http://mathoverflow.net/questions/106480/which-polynomials-are-solvable-by-radicals Comment by pritam pritam 2012-09-06T06:59:51Z 2012-09-06T06:59:51Z Please see the edit, I am interested in some simple characterization for some special values of $n$ if possible. http://mathoverflow.net/questions/105907/why-is-the-physical-space-equivalent-to-mathbbr3 Comment by pritam pritam 2012-08-30T07:47:28Z 2012-08-30T07:47:28Z Isn't the topological structure inherited from its algebraic structure, I mean the metric on $\mathbb{R}$ is $|a-b|$ which is defined according to its algebraic structure http://mathoverflow.net/questions/105907/why-is-the-physical-space-equivalent-to-mathbbr3 Comment by pritam pritam 2012-08-30T07:29:45Z 2012-08-30T07:29:45Z @Mariano: But mathematicians also use this fact quite often, to represent real numbers we intuitively assume they are lying on a straight line (say, drawn on a piece of paper). http://mathoverflow.net/questions/105907/why-is-the-physical-space-equivalent-to-mathbbr3 Comment by pritam pritam 2012-08-30T07:21:51Z 2012-08-30T07:21:51Z @Damien: I also thought people may find it off-topic, what do you mean by a real question ? http://mathoverflow.net/questions/105543/algorithms-to-find-irreducible-polynomials-of-a-given-degree/105558#105558 Comment by pritam pritam 2012-08-30T06:03:31Z 2012-08-30T06:03:31Z @Igor Rivin: Thanks for the link, I was more interested in the second question. http://mathoverflow.net/questions/105543/algorithms-to-find-irreducible-polynomials-of-a-given-degree Comment by pritam pritam 2012-08-26T16:35:57Z 2012-08-26T16:35:57Z Extremely sorry for the typo, I was confusing with $\mathbb{F}_p$ http://mathoverflow.net/questions/105299/diametrically-opposite-points-go-to-diametrically-opposite-points-under-stereogra/105301#105301 Comment by pritam pritam 2012-08-23T06:56:10Z 2012-08-23T06:56:10Z I dont get the proof for the first implication, I think they are considering two antipodal points on the Riemann Sphere and $z$ and $z^\prime$ are their projections. I know the second implication. http://mathoverflow.net/questions/105299/diametrically-opposite-points-go-to-diametrically-opposite-points-under-stereogra Comment by pritam pritam 2012-08-23T06:40:19Z 2012-08-23T06:40:19Z I meant circle centred at any arbitrary point, is it really false then ? Basically I want a proper proof without using geometric intution. http://mathoverflow.net/questions/104504/change-in-angle-between-curves-due-to-stereographic-projection Comment by pritam pritam 2012-08-11T17:54:37Z 2012-08-11T17:54:37Z This is not a homework, this question came up in my mind while studying stereographic projections. Please give me some hint so that I can proceed. http://mathoverflow.net/questions/100343/how-many-ways-are-there-to-define-additions-and-multiplications-in-the-ring-of-in Comment by pritam pritam 2012-06-22T12:26:59Z 2012-06-22T12:26:59Z the second case seems interesting can we say something ? http://mathoverflow.net/questions/99185/a-ring-of-fractions-which-has-finitely-many-maximal-ideals Comment by pritam pritam 2012-06-11T09:27:48Z 2012-06-11T09:27:48Z @Angelo: This is not a homework problem. I dont have any homework problem in my math courses.