# 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.

46 views

### $\mathbb{Z}_p$ is an Integral Domain

Assume that we define the ring of the $p$-adic integers as the projective limit $$\mathbb{Z}_p =\varprojlim \frac{\mathbb{Z}}{p^n\mathbb{Z}}$$ Then $\mathbb{Q}_p$, the field of the $p$-adic numbers ...
26 views

### Given a homomorphism defined on a generating set of a group how to define it for a general element?

Let $\mathbb Z^n$ and $\mathbb Z^d$ be free $\mathbb Z$-modules with $d>n$. Suppose $v_1,\dots,v_d$ are primitive vectors in $\mathbb Z^n$. Let $e_1^*,\dots, e^*_n$ be the dual basis (that is, a ...
81 views

### Number of subgroups equal to order of group

Here is a fun question. Consider the dihedral group $\mathcal{D}_4=\left\langle a,b\mid a^4=b^2=1, bab=a^{-1}\right\rangle$ of order $8$. This group has exactly $8$ genuine subgroups (but not all ...
34 views

### Are there groups $G$ isomorphic to $\mathrm{Aut}\left( \mathrm{Aut} (G)\right)$, such that $G\not\cong \mathrm{Aut}(G)$?

Trivial examples for the first condition are easy to find: $G={1},C_2$. Are there any (finite\non-finite) groups that satisfy the conditions in the title?
47 views

77 views

### Can the dihedral groups be seen as subgroups of each other?

I am solving one difficult problem and now I need information on this: If $m\mid n$, is then possible to "embed" $D_m$ to $D_n$, or otherwise said, does $D_n$ have a subgroup isomorphic to $D_m$? ...
261 views

### Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
41 views

### Quotient ring with reducible polynomial

Let $S = \mathbb{R}[x]/(x^2+1)^2).$ The first goal is to show that there exist exactly two homomorphisms $\pi\colon S\to \mathbb{C}$ such that $\pi|_{\mathbb{R}} = \text{id}_{\mathbb{R}}.$ I know ...
49 views

### Factoring $p(x) = x^n -1$ for any natural number $n$

Can I say that from inspection, $(x-1)$ is a factor which implies that $p(1) = 0$. I then used long division which then gave this: $p(x) = (x-1)(\sum \limits _{i=1} ^{n} x^{n-i})$ How would you have ...
45 views

I am confused about this notation from the image below. How can you represent something like (1,2),1 as a 2D matrix? For S1 = (0,1,2) and S2 = (0,1) I would expect two matrices like: [(1,0),(1,0),...
56 views

### $17\mathbb{Z}[\sqrt {10}]$ is prime ideal in $\mathbb{Z}[\sqrt {10}]$

This seems tedious since I would have to show $(a+b\sqrt {10})(c+d\sqrt {10})=17k+17j\sqrt {10}$ implies that one of the factors belongs to $17\mathbb{Z}[\sqrt {10}]$. It's easier to show something ...
69 views

55 views

### Examples of irreducible polynomials over a finite field with prescribed coefficients [on hold]

I came to know that it is an open problem, but I am not able to find any simple example to explain it properly. Can some one help me with some simple explanation regarding what this problem is about?...
113 views

### Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
32 views

### $\mathbb{Z}[\sqrt{-5}]$ satisfies the descending chain condition of divisors

I want to show that $\mathbb{Z}[\sqrt{-5}]$ satisfies the descending chain condition of divisors: given a chain $a_1,a_2,\dots,a_n,\dots$ and $a_{n+1}\mid a_n$ for any $n\in \mathbb{N}$, then there is ...
47 views

### $I$ is the maximal left ideal

Let $R$ be a ring and $I\subseteq R$ the unique maximal right ideal of $R$. I have shown that $I$ is an ideal and that each element $a\in R-I$ is invertible. I want to show that $I$ is the unique ...
16 views

### Let $M$ be a finitely generated module over a P.I.D.. If $M=A\oplus B$, $A$ is a torsion submodule and $B$ is free, then $A=M_{\text{tor}}$.

I am reading the Goodman's Algebra. There is a lemma. Let $M$ be a finitely generated module over a principal ideal domain $R$. (a) If $M=A\oplus B$, $A$ is a torsion submodule, and $B$ is free, ...
64 views

### Is $a$ invertible?

We have that $R$ is a ring. Suppose that $Ra=R$ and $bR=R$, for $a,b\in R$. Then we have that there is $x\in R$ such that $ab=1$ and $bx=1$. Does it follow from that that $a$ is invertible?
57 views

### Cardinallity increasing constructions

If we start with $\mathbb{Z}$ we can through localization get $\mathbb{Q}$, but that has the same cardinallity as $\mathbb{Z}$, so it doesn't increase cardinality for infinite sets, which is what I am ...
42 views

### Is an algebra homomorphism between two finitely generated algebras over a field automatically an integral morphism?

I'm having a bit of trouble with the idea of an integral morphisms, and algebra homomorphisms for that matter. I'm wondering if the above is just "automatically" true. Does an algebra over a field ...
40 views

### What are the invertible elements of the $K[X]$ ring where $K$ is a field? [duplicate]

What are the invertible elements of the $K[X]$ ring where $K$ is a field? I know that in a field all the elements are invertible but I'm having trouble linking $K$ and $K[X]$
65 views

### Image drawing complex analysis [on hold]

$w=u+iv,z=x+iy$ are complex numbers and we have $w=z^2-2z$. Determine the image in the $w$-plane of the unit circle $x^2+y^2=1$. I have tried to answer this here Question and Answer. I have problems ...
61 views

We know that $x^2 -3$ is irreducible in $\mathbb{Q}[x]$. We also know that $\sqrt{3}$ solves $x^2 - 3$. As a result, $x^2 - 3 = (x-\sqrt{3})(x+\sqrt{3})$. This implies that the splitting field of $x^2 ... 2answers 64 views ### If$R\otimes_\mathbb R\mathbb C$is finitely generated$\mathbb C$- algebra then$R$is a finitely generated$\mathbb R$- algebra? Let$R$be an$\mathbb R$- algebra. Suppose$A=R\otimes_\mathbb R\mathbb C$is a finitely generated$\mathbb C$- algebra then is$R$a finitely generated$\mathbb R$- algebra? I thought along the ... 0answers 77 views ### Simple examples of rings from topology The ring$C([0,1],\mathbb{R})$of continuous functions from$[0,1]$to$\mathbb{R}$is an interesting example of ring due to its some interesting property (namely, structure of maximal ideals). The ... 0answers 44 views ### Field of fractions of ring F[x] [closed] Let$F$be a commutative ring without zero divisors and$Q$its field of fractions. Prove that field$Q(x)$is a field of fractions of ring$F[x]$Thanks for any help. 2answers 63 views ### Legendre symbol$(-21/p)$I am a bit confused with the question: For what prime$p$,$\left(\frac{-21}{p}\right) = 1$? I did something like that: $$\left(\frac{-21}{p}\right) = \left(\frac{-1}{p}\right)\left(\frac{3}{p}... 0answers 57 views ### Which functions \mathbb{Z} \rightarrow \mathbb{Z} are 'totally compatible'? Definition 0. For each integer k and each function f : \mathbb{Z} \rightarrow \mathbb{Z}, lets define that f is k-compatible iff there exists a function g : \mathbb{Z}/k\mathbb{Z} \rightarrow ... 1answer 35 views ### In transitive (non-trivial) group action, there must be at least one group element without fixed point Let a finite group G act transitively on a finite set S with |S| \geq 2. The problem is to show that not every g \in G can have a fixed point in this action. I proved this on my own, but I'm ... 0answers 30 views ### prove Euclidean ℚ^p ℚ_p [closed] prove Euclidean 1) ℚ^p with norm n(kp^m) = |k|, k,m ∈ ℤ, (k, p) = 1; 2) ℚ_p with norm n(\frac{a}{b}p^m) = p^m, a,b,m ∈ ℤ, m≥0 (ab, p) = 1; 1answer 28 views ### Find the minimum, maximum, minimals and maximals of this relation Tell if the following order relation is total and find the minimum, maximum, minimals and maximals:$$\forall a,b \in\mathbb Z,\ \ a\ \rho\ b \iff a \leq b\ \text{ and }\ \pi(a) \subseteq \pi(b)$$... 2answers 82 views ### How to show \sqrt{3} + \sqrt{2} is algebraic? [duplicate] How to show \sqrt{3} + \sqrt{2} is algebraic? Is there a way I can do this without trial and error? Thanks. 1answer 35 views ### Algebra problem about Ker and Im I have a problem with this linear algebra exercise. A) Find an orthonormal basis with respect upon the Euclidean product for a vector space in \mathbb{R}^3, generated by those vectors: (1, 2, -1)... 1answer 26 views ### Interpretation of the join of two stabilizer subgroups Let G be the group acting on two sets X, Y. Let G_x and G_y be stabilizer subgroups of some elements x \in X, y \in Y. It is easy to see that G_x \cap G_y = G_{(x,y)}, when we combine two ... 2answers 33 views ### Which one of these two is an equivalence relation I'm having an issue with the following exercise: Given \alpha and \beta two binary relationships defined in Z such that:$$\forall m,n \in Z, n\ \alpha\ m \iff n = m\ \ \vee\ \ rest(n,7)\ +... 3answers 1k views ### A binary operation, closed over the reals, that is associative, but not commutative I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ... 2answers 35 views ### If$a$has order$p$and$aPa^{-1}=P$then$a\in P$If$G$is a finite group,$P$a$p$-Sylow subgroup of$G$,$a\in G$has order$p$and$aPa^{-1}=P$then$a\in P$. This is proved in Rotman: Proof: We have$a\in N_G(P)$. If$a\notin P$then$aP\in ...
let $p$ be a prime number; then how many distict subrings (with unity) of cardinality $p$ does the field $F_{p^2}$ have? 1.$0$ 2.$1$ 3.$p$ 4.$p^2$ I think it will have only one subring of ...