# Tagged Questions

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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### Finding dimension of a field extension

How would anyone go about this problem? Find dim$_\mathbb{Q}\mathbb{Q}(\alpha,\beta)$ where $\alpha^{3}=2$ and $\beta^{2}=2$. Thanks for your help, I really don't know how to go about this ...
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### Verifying orthogonality between two binary sequences

I have studied that for orthogonality to exist between two binary sequences: [Number of bit agreements - Number of bit disagreements]/sequence length=0 Eg, for an orthogonal matrix X given by: ...
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I came across something related to the degree of a splitting field for a polynomial over a field $K$. Let's suppose $p \in K[x]$ with degree $n$, and $p$ has irreducible factors $f_{1}, \ldots, ... 1answer 20 views ### How to show that$\prod^{p-1}\limits_{j=0} (x+\eta^jy)=x^p+y^p$How to show that$\prod^{p-1}\limits_{j=0} (x+\eta^jy)=x^p+y^p$, where$p$is an odd prime,$\eta$is$p$-th roots of unity and$x,y$are integers. It could be reduced to the form ... 0answers 11 views ### If$g=(n-j,\dots,n)$and$\sigma\in S_{n-1}$, why are the inversions of$g\sigma$the union of the inversions of$\sigma$and$g$? I can't see why a claim I'm reading is true. If$\sigma\in S_n$, let$R(\sigma)=\{(i,j):i<j,\ \sigma(i)>\sigma(j)\}$, i.e.,$R(\sigma)$is the set of inversions of$\sigma$. The set ... 0answers 23 views ### Why do I get homogenizations of polynomials by trying to find roots in$\mathbb Q$. I noticed that if I have a polynomial equation in, say$x$that needs to be solved in$\mathbb Q$, one tactic is to substitute$x=y/z$where$y$and$z$are coprime integers, but then after clearing ... 0answers 23 views ### Smoothness in cyclotomic versus complex fields? Say we have a polynomial in a cyclotomic field; in particular, an n-th cyclotomic field, where n is the order of the polynomial's symmetry group. If we know the polynomial is smooth over this field, ... 1answer 34 views ### Is there any group-like structure that doesn't have an identity, but has (non-equal) left and right identities? Is there any group-like structure that doesn't have an identity, but has a (non-equal) left and right identities? 1answer 31 views ### Representation of$(\mathbb{Z}_{\frac{*}{5}},_{\times5})$using Cayley table [on hold] Could someone give a hint how to represent group$(\mathbb{Z}_{\frac{*}{5}},_{\times5})$using Cayley table? Thanks for replies. 0answers 39 views ### Conjecture on a graded ring Consider$B^{(n)}=\mathbb{F}_2[X_1,\dots,X_n]/(X_1^2,\dots,X_n^2)=\bigoplus_{i=0}^nB_i^{(n)}$, where$B_i^{(n)}$is the space of homogeneous elements of degree$i$. Notice that ... 1answer 46 views ### Cancel an element that's not a unit I was going through the proof of "every PID is a UFD" in Serge Lang's Algebra-book and something confused me. When it comes to proving the uniqueness of the factorization, he writes: "Suppose$a$... 0answers 17 views ### Isomorphism Between Splitting Fields Let$f \in F[x]$be irreducible and split in$E/F$. Suppose that$F \cong F'$via$\phi$, and let$f' \in F'[x]$be polynomial obtained by applying$\phi$to the coefficients of$f$. It can be shown ... 1answer 24 views ### What is the size of Range? Suppose d=gcd$(a,n)$where$a, n \in \mathbb{Z}, n>0$and$f_a: \mathbb{Z_n} \to \mathbb{Z_n} \\ \qquad x \to ax\texttt{ mod } n$The size of Domain is evident and for the size of Range my ... 0answers 21 views ### Milnor patching for general modules The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings $$\begin{array}{} R & \xrightarrow{f_2} & R_2 \\ \downarrow{f_1} & ... 0answers 23 views ### Derivations of algebra matrices? I see on internet that all derivations of algebra of matrices M_n(\mathbb{R}) with respect to it commutator or matrices multiplication, are inner. I do not know how to prove this fact. Any hint are ... 1answer 27 views ### Expressing polynomial as linear combinaion I found these questions in Adams Introduction to Groebner bases Let f=x^6-1 and g=x^4+2x^3+2x^2-2x-3. Let I=\langle f,g\rangle. Calculate the polynomial that generates I alone. After a ... 1answer 49 views ### Maximal Ideals of \mathbb C[x, y] I recently learnt that the maximal ideals of \mathbb C[x, y] are precisely the ones of the form (x-a, y-b) for some a, b\in \mathbb C. I am unable to prove it. So I considered an easier ... 0answers 31 views ### Intuition on S_4/K_4\cong S_3 [duplicate] This is a question I was pondering on my way to class, and may not have an answer. Take the normal copy of K_4 in S_4 (The group K_4 \cong\{e,(12)(34),(13)(24),(14)(23)\}, not the non-normal ... 1answer 22 views ### Induction and Compact induction of representations Let H \leq G be a subgroup of a finite group, G. Suppose (\sigma, W) is a representation of H. Then we know that Ind_H^G \sigma and ind_H^G \sigma are isomorphic, where$$Ind_H^G ... 1answer 41 views ### Localization of a coherent module is coherent I would like to prove that the coherence of a sheaf on a scheme is affine-local. For this, it is necessary to prove that for a coherent$A$-module$M$, its localization$M_{f}$at$f\in A$is a ... 4answers 135 views ### Is there any neat way to show$\phi$is a homomorphism? In Michael Artin's Algebra (chapter 2, page 50, example 2.5.13) the author illustrates a homomorphism from$S_4$(all permutations of indices$(1,2,3,4)$) to$S_3$(all permutations of indices ... 3answers 32 views ### Number of left cosets equals number of right cosets? So I've been working on abstract algebra out of John B. Fraleigh's 3rd edition text. In the exercises of chapter 11, I came upon a question which I cannot even begin to solve. "Show that there are ... 1answer 77 views ### Why would we a priori expect$V(I)$to satisfy axioms to define the closed sets for a topology on$\text{Proj}(S)$? The topological space$\text{Proj}(S)$has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq \mathfrak{p}\},$$and the closed sets are the loci ... 1answer 20 views ### For a root system, why does$\beta\in\Delta_+\setminus\{\alpha_i\}$imply$(\beta+\mathbb{Z}\alpha_i)\cap\Delta\subset\Delta_+$? Let$\mathfrak{g}(A)$be a Kac-Moody algebra for a matrix$A$, with root basis$\{\alpha_1,\dots,\alpha_n\}$. There is a remark on the bottom of page 6 of Kac's Infinite Dimensional Lie Algebras ... 0answers 31 views ### Why is$\mathfrak{g}(A)=\mathfrak{g}'(A)$iff$\det(A)\neq 0$? In many sources (Victor Kac, Zhexian Wan, etc.), it's stated as a remark that if$\mathfrak{g}(A)$is the Kac-Moody algebra of a generalized Cartan matrix$A$, then$\mathfrak{g}(A)=\mathfrak{g}'(A)$, ... 2answers 48 views ### How do we conclude that the relation is equal to$1$? We have a curve of the form $$s^2-\alpha t^2=1 \tag 1$$ (in$ts$-coordinates). If$(a,b)$and$(c,d)$are points of$(1)$, then$(ac+\alpha cd, ad+bc)$is point of$(1)$.$(a,b)$is a point of ... 1answer 35 views ###$R$is a unique factorization domain$\iff$every prime minimal over a principal ideal is also principal I'm trying to show that a ring$R$is a unique factorization domain$\iff$every prime minimal over a principal ideal is also principal. I think the idea is to use the principal ideal theorem of ... 2answers 66 views ### Commutative binary operations on$\Bbb C$that distribute over both multiplication and addition Does there exist a non-trivial commutative binary operation on$\Bbb C$that distributes over both multiplication and addition? In other words, if our operation is denoted by$\odot$, then I want the ... 0answers 20 views ### Show that every algebraic field extension has a normal closure I am trying to show that every algebraic extension of fields has a normal closure. Let$L/K$be an algebraic extension. First suppose that the extension is finite. So$L=K(\alpha_1,...,\alpha_n)$... 0answers 51 views ### Is there a polynomial$p$such that it is bijective and$ p: \mathbb{Q}^n \rightarrow \mathbb{Q}$for$ n>1$?? Let us define a polynomial$p$defined as follow $$p: \mathbb{Q}^n \rightarrow \mathbb{Q}.$$ I acrossed this question in mind. Is there a polynomial$p$such that it is bijective and$p: ...
$Z_{5}$ is the group to act on the set $\{ 1,2,3,4,5\}$. In how many ways is that possible $?$ Now $0$ will give the identity map. $1$ will give a bijection in $5!$ ways so ...