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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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 ...
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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: $$\forall a,b \in\mathbb Z,\ \ a\ \rho\ b \iff a \leq b\ \text{ and }\ \pi(a) \subseteq \pi(b)$$ ...
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83 views

Factor ring and prime elements

My task is to find prime elements of the ring $\mathbb Z[\sqrt{-21}]$ and describe the factor ring $\mathbb Z[\sqrt{-21}]/(2+\sqrt{-21})$. I think that to describe factor-ring i need to find the ...
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1answer
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 $K[X]$
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1answer
3k views

Units of polynomial rings over a field

If $R$ is a field, what are the units of $R[X]$? My attempt: Let $f,g \in R[X]$ and $f(X)g(X)=1$. Then the only solution for the equation is both $f,g \in {R}$. So $U(R[X])=R$, exclude zero elements ...
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2answers
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Finding invertible polynomials in polynomial ring $\mathbb{Z}_{n}[x]$ [duplicate]

Is there a method to find the units in $\mathbb{Z}_{n}[x]$? For instance, take $\mathbb{Z}_{4}$. How do we find all invertible polynomials in $\mathbb{Z}_{4}[x]$? Clearly $2x+1$ is one. What about ...
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1answer
65 views

Image drawing complex analysis [closed]

$w=u+iv,z=x+iy$ 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 ...
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61 views

Find a splitting field of $x^2 + 1$ over $\mathbb{Z}_3$

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 ...
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Help to prove (decomposition of gcd) [closed]

Let $K$ be a unique factorization domain (factorial ring) and $Q$ its field of fractions. Prove that for each $f, g \in K[x]$ $$ (f, g)=(d(f), d(g))(\alpha(f), \alpha(g)),$$ where G.C.D. of content $...
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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 ...
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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 ...
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1answer
84 views

Has the following algebraic structure been studied before

I have recently been thinking about the following mathematical structure. I am wondering if this structure has been studied before or if it is new (I doubt it as it seems a fairly obvious structure ...
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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.
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Non-algebraic structures?

We call group, ring, field,... "algebraic structures". Do we have similar analogue for transcendental numbers? If not, then how do we study interactions between various transcendental numbers? Also, ...
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80 views

Are there infinitely many pairs of primes, $p$ and $q$, such that $q = 4p + 1$?

How close can one come to proving that there are infinitely many primes, $p$ and $q$, such that $q = 4p + 1$? The idea for this question came from reading the question and answers posed by user39898,...
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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 ...
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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 ...
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2answers
42 views

What is the minimum and maximum of a set with only one element?

This is surely a trivial question but I want to be sure I understand correctly what happens. Given a set $A = \{1\}$, what is $\min A$ and $\max A$? Is it $\min A = 1$ and $\max A = 1$?
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4answers
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polynomial ring with isomorphic quotients

If $R$ is a commutative ring and $f(x), g(x) \in R[x]$ two polynomials such that $R[x]/f(x)\cong R[x]/g(x)$ as $R$-algebras, what can we say about $f$ and $g$? Or given $f(x)\in R[x]$, what can we ...
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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 $n(\frac{a}{b}p^m) = p^m$, $a,b,m ∈ ℤ,$ $m≥0$ $(ab, p) = 1$;
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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.
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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: $(1, 2, -1)$...
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Precedence of operations in vector spaces

Suppose that $V$ is a vector space over $\mathbb R$ (for simplicity) with addition denoted by $\oplus$ and scalar multiplication denoted by $\otimes$. Let $\mathbf u, \mathbf w \in V$ and let $\lambda ...
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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 ...
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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)\ +...
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3answers
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A question on identifying normal subgroups of $S_4$

Let $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(12)(34),(13)(24),(14)(23)\}$ be subgroups of $S_4$, where $e$ denotes the identity element of $S_4$. Which of the following are correct? $H$ and $K$ are normal ...
4
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107 views

Prove that a specific ring of integers is not monogenic

I'm trying to prove that the ring of integers of $K=\mathbb{Q}(\sqrt7, \sqrt13)$ is not of the form $ \mathbb {Z}[a]$ for some $a$. Unfortunately I can not figure out where to start. I tried to ...
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62 views

Subfields of $\mathbb{C}$ which are connected with induced topology

The ring of continuous functions on $[0,1]$ to $\mathbb{R}$ has an interesting property: every maximal ideal of this ring is the subset of all functions vanishing at a common point. If we ...
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Finding properties of operation defined by $x⊕y=\frac{1}{\frac{1}{x}+\frac{1}{y}}$? (“Reciprocal addition” common for parallel resistors)

I have recently found some interesting properties of the function/operation: $x⊕y = \frac{1}{\frac{1}{x}+\frac{1}{y}} = \frac{xy}{x+y}$ where $x,y\ne0$. and similarly, its inverse operation: $x⊖y = ...
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1answer
89 views

Number of subrings of a Field

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 ...
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33 views

Different right / left identity and two sided identity element

Maybe this will be a trivial question but I can't find a solution for this. I checked already on a lot of questions here and on google but can't find a solution for my specific problem. Given an ...
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1answer
26 views

Question about multiplicative arithmetic functions

Let $f,g:\Bbb N\to \Bbb C$ be multiplicative arithmetic functions, i.e. $$\gcd(m,n)=1\implies f(mn)=f(m)f(n)$$ and same for $g$. We can also assume $f(1)\neq 0$ and $g(1)\neq 0$ if necessary. How can ...
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1answer
38 views

Showing an isomorphism of rings

Consider the ideal $I=(1+2x)\cdot \Bbb Z[x]$ in the polynomial ring $\Bbb Z[x]$. I am trying to show that $\Bbb Z[x]/I$ is isomorphic to $R=\{\frac{a}{2^r}:a\in \Bbb Z, r\in \Bbb N_0\}$. My approach:...
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1answer
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Homomorphism, Kernel and Image

Let $f$ the homomorphism $f: (\mathbb{Z},+)\to S_6, f(1)=(123)(456).$ Find the Kernel and image of $f$. Hello, my question is how solve this $(123)(456)$? In form matrix, because can to be $(123)(456)...
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Frobenius-Perron dimension on a fusion category

Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$. The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is ...
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Is Tambara-Yamagami category admits a braiding when G is a nonabelian group?

Tambara-Yamagami category is a fusion category which its simple objects are elements of a group and one element out of group. i.e : $$simple\;objects = G \cup \{m\}$$ The fusion rule of this ...
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Question about center of a group/subgroup.

Let $G$ a non abelian group of order $p^aq^b$ where $p,q$ are prime and $p^aq^b$ is not prime. I want to show that $G$ is not simple. In a proof of my course, they do as following for the case where ...
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1answer
24 views

Finitely generated Ideals of finite algebras

i would really appreciate any help with this question. So, the question is: How to prove that finitely generated ideals of finite algebras over the ring F are finite over F? Thanks.
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1answer
404 views

Proving that a group of order $pqr$ (with conditions on those primes) is abelian.

I'm doing this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Also suppose that $p \not\equiv 1$ (mod $r$), $p \not\equiv ...
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What happens if I take a quotient over a reducible polynomial?

I know that for adjoinging roots to a field, I need to find irreducible polynomials so that the ideal I am taking the quotion with will be maximal, hence the resut being a field. Imagine I am working ...
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Nonprincipal lattice

Let $\Lambda$ be a $J$-module generated by the elements $v_1, \ldots, v_n$ which are linearly independent over $\mathbb{C}$, $v_i \in \mathbb{C}^n$. It is said that in case $J$ is not a PID (for ...
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1answer
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prove that $x^{2\cdot 3^n}+x^{3^n}+1$ is not primitive (mod 2)

I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, It seems that there are no primitive polynomial of the form $$x^{2k}+x^{k}+1\quad k>1$$ and I'...
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3answers
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Quick question about $\mathbb{C}$ considered as field extension of $\mathbb{R}$

In algebra, one way to understand $\mathbb{C}$ is to consider it as a field extension of $\mathbb{R}$. What (sometimes) worries me is this: From this point of view, does $\mathbb{C}$ really contain a ...
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Does Nagata theorem hold in a field that is not algebraically closed?

Let $R$ be a finitely generated $k$ - algebra and $G$ be a reductive group acting rationally on it. Then a theorem of Nagata says that the invariant ring $R^G$ is also finitely generated. Here $k$ ...
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Let $\vert G \vert = p^n m$ where $m < 2p$. Show that $G$ has a normal subgroup of order $p^{n-1}$ or $p^n$

Let $G$ have order $p^n m$ where p is a prime and $p \nmid m$. Suppose $m < 2p$. Show that $G$ has a normal subgroup of order $p^{n-1}$ or $p^n$. I have tried to apply the Sylow Theorems but I ...
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22 views

Reducible/irreducible polynomial [duplicate]

Prove that polynomial $ x^{p^m}-\alpha \in k[x], chark = p > 0 $ irreducible over $k$ or $ \exists\beta\in k, \alpha = \beta^p $.
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1answer
48 views

Can I deduce $f(\textrm{Ker}(g))=g(\textrm{Ker}(f))=0$ from this data?

Let $R$ be a commutative ring with identity and $A, B$ two $R$-algebras. Consider $f, g: A\longrightarrow B$, $h:B\longrightarrow A$ and $\imath:A\longrightarrow A$ morphisms of $R$-algebras ...
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2answers
80 views

Minimal polynomial of $\alpha = \cos\left(\frac{\pi}{48}\right)$ over $\mathbb Q$

This is a homework problem, so just a nudge in the right direction would be great. So I am required to show that $\alpha$ is a algebraic over $\mathbb Q$ and show that the degree of its minimal ...
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0answers
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$R$-module strcture on $M\oplus A$ so that it becomes a $R$-algebra?

Let $R$ be a commutative ring with identity, $A$ an associative $R$-algebra without identity and $M$ a $A$-bimodule. I read I can endow $M\oplus A$ with a structure of $R$-algebra with the product $$(...