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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What are $S_{n}$ and $A_{n}$ in group theory?

What are $S_{n}$ and $A_{n}$ in group theory, and is $[S_{4},A_{4}]=4$? I know that $S$ has to do with permutations, but I am not sure if thats right. Thanks,
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1answer
23 views

Abstract Algebra: Proof involving a p-Sylow, nomal subgroup P in G.

Let $P$ be a normal subgroup in $G$ where $P$ is a $p$-Sylow subgroup of $G$. Show that $\phi(P)=P$ for every automorphism $\phi$ of $G$.
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19 views

Homomorphisms between abelian groups

Let $H$ and $K$ be finite abelian groups. Assume that $H$ has order $n$ and $K$ has order $m$. Suppose that $n$ and $m$ are relatively prime. This implies that any homomorphism $\alpha : H\rightarrow ...
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2answers
25 views

If $M\bigoplus N $ submodule $A\bigoplus B$ does it imply either $M$ submodule $A$ or $M$ submodule $ B$

If $M\bigoplus N$ is a submodule of $A\bigoplus B$ does it necessarily imply either $M$ is a submodule of $A$ or $M$ is a submodule of $B$. If in general it is not true, is it true if $M$ and $N$ ...
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does dicyclic group of order $2^{\alpha}m$, $(2,m)=1$ have normal subgroup of order $m$?

consider dicyclic group with presentation $$Dic_n=< a,b |a^{2n}=1,a^n=b^2,ba=a^{-1}b>$$ the order of this group is $4n$,suppose that $|Dic_n|=2^{\alpha}m$ where $(2,m)=1$ my question is about ...
3
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2answers
50 views

Is $\mathbb{Z}[2\sqrt{2}]$ a PID?

I am practicing for my algebra qual and I would like to know if $\mathbb{Z}[2\sqrt{2}]$ is a PID. I had no intuition at first except the fact that $\mathbb{Z}[i\sqrt{2}]$ is a ED with norm ...
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2answers
25 views

How do I compute the norm of a non-principal ideal of the ring of integers of a quadratic field without using ''large'' results

I am trying to compute the norm of the ideal $I=(7, 1+\sqrt{15}) \trianglelefteq \mathbb Z[\sqrt{15}],$ the ring of integers of $\mathbb Q[\sqrt{15}].$ I knew $I^2$ would be principal, as $I\bar ...
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56 views

Consider extension $[\mathbb{Q}(\alpha\ ):\mathbb{Q}]$ where $\alpha\ $ is zero of $P(x) = x^4 + 9x^{2} + 15 $.

Consider extension $[\mathbb{Q}(\alpha):\mathbb{Q}]$ where $\alpha$ is zero of $p(x) = x^4 + 9x^{2} + 15 $. Find $[\mathbb{Q}(\alpha):\mathbb{Q}(\alpha^2 + 3)]$. My attempt: By Eisenstein's ...
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19 views

Newton method for $p$-adic fields

I want to understand where the last line comes from. I.e. why there is the $p^{2n-2ka}$ term. I tried to use the estimate formula for the reminder but it doesn't work for me...
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1answer
39 views

$M$ is a faithful $A$-module.

Let $A = M_2(\mathbb{C})$ and let $M = \mathbb{C}^2$ with its usual $A$-module structure. How do I show that $M$ is a faithful $A$-module? I understand that this amounts to showing that its ...
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3answers
47 views

Does there exist a module structure over $\mathbb{C}^2$ as a $\mathbb{C}[x]$-module?

Let $R = \mathbb{C}[x]$, $M = \mathbb{C}^2$. Does there exist a module structure over $M$ as an $R$-module or not?
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2answers
20 views

Module $M$ is infinite dimensional as a $\mathbb{C}$-vector space.

Let $M$ be a $\mathbb{C}[x]$-module. Since $\mathbb{C} \subset \mathbb{C}[x]$, we may consider $M$ as a $\mathbb{C}$-vector space. If $M$ is faithful, must $M$ be infinite dimensional as a ...
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1answer
36 views

Determine if $(((13)),\circ)$ is a normal subgroup of $(S_{3},\circ)$ [on hold]

Let $((13))$ denote the group generated by $(13)$. Is $(((13)),\circ)$ a normal subgroup of $(S_{3},\circ)$? Also is $(((123)),\circ)$ a normal subgroup of $(S_{3},\circ)$? I have just started ...
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1answer
36 views

Equivalence between category of $R$-modules and $S$-modules

Is it true that by equivalence between category of $R$-modules and $S$-modules we conclude that $R$ and $S$ are isomorphic?($R$ and $S$ are unital rings.)
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Exists an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?

Let $M$ be an $R$-module with some torsion element $m$. Does there exist an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?
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1answer
38 views

all of subgroups of group

Is the way to gust that in finite group how many subgroup of same order?I ask this question because when draw the lattice diagram of subgroups of group sure that all of them describe. Thanks for hint
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1answer
26 views

Arbitrary elements in a quotient ring $\Bbb R[x]/(x-1)$

If I have an ideal $(x-1)$ for the ring $\Bbb R[x]$, how do I think of the quotient ring $\Bbb R[x]/(x-1)$? I have all polynomials with: $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0 {\pmod {x-1}}$$ ...
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1answer
30 views

Properties of homomorphisms

I have some problems in how to prove these: Let $f$ be homomorphism from group $G$ to a group $N$. Prove the following: $k\le G$ iff $f[k]\le N$ $f$ is onto iff range of $f =N$ $f$ is one-to-one ...
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29 views

Heyting algebras and infinite distributive law

I want to prove that "a complete lattice satisfies the infinite distributive law $a\wedge(\vee{S})=\vee\{a\wedge s|s\in S\}$ iff it is a Heyting algebra". I proved "if" part but can't "only if" part. ...
5
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1answer
32 views

Group ring of a cyclic group over a finite field

Suppose $ p $ a prime integer and $ n $ a positive integer. Does anyone know off the top of their heads if the group ring $ \mathbb{F}_{p}[\mathbb{Z}/n] $ (perhaps regarding $ \mathbb{Z}/n $ as the $ ...
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Name for the reals augmented with an $x$ such that $x^2 = x$

If you add an $x$ such that $x^2=-1$ to the reals, you get the complex numbers. If you add an $x$ such that $x^2=0$ to the reals, you get the dual numbers. If you add an $x$ such that $x^2=+1$ to the ...
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4answers
69 views

Prove that $a$ and $a^{-1}$ inverse have the same order in $Z_n$

So there is a question in my lecture notes that I'm not too sure how to approach. It reads as follows: Suppose $a$ is invertible modulo $n$. Prove that $a$ and $a^{-1}$ have the same order in ...
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1answer
24 views

Field Theory Problem in Beachy's Abstract Algebra involving field extensions and transcendental elements.

Let $\mathbb{F}=\mathbb{K}[u]$ where u is transcendental over $\mathbb{K}$. Show that if $\mathbb{K} \subsetneq \mathbb{E} \subseteq \mathbb{F}$ then u is algebraic over $E$. I'm guessing that I need ...
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1answer
17 views

Let $K\subseteq F$ be a finite extension of fields, $M$ and $N$ intermediate fields separable on $K$. Show that $MN$ (compositum) is separable.

Let $K\subseteq F$ be a finite extension of fields, $M$ and $N$ intermediate fields separable on $K$. Show that $MN$ (compositum) is separable. Using the Element Primitive Theorem, we know that ...
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1answer
29 views

Group of order 10 has an element of order 5, without using Cauchy's or Sylow's theorems

This is almost a duplicate of the following questions (but, read further): Group of order $63$ has an element of order $3$, without using Cauchy's or Sylow's theorems Show any group of order ...
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1answer
52 views

Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. [duplicate]

Homework question: Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. Can you give me some hints ...
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1answer
35 views

about left identity in a ring..

Let $S$ be the subset of $\mathbb{M}(\mathbb{R})$ consisting if all matrices of the form : $\begin{pmatrix} a & a \\ b & b \end{pmatrix}$ The matrix $\begin{pmatrix} x & x \\ y & y ...
4
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38 views

Ideals in a quotient

I am interested in finding the number of prime ideals in $\mathbb{Z}[x]/(12,x^2+1)$. Here is what I think. Modding out by $x^2+1$, we get $\mathbb{Z}[i]/(12)$. Factoring $12$ in the Gaussian integers, ...
3
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2answers
43 views

Understanding Quotient Rings

I am watching a video on Field Extensions (trying to self "relearn" some Abstract Algebra before I take it again). I struggled with it as an undergrad, so I'm trying to get a leg up. The example ...
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A question about a property of Gauss sum.

I am reading the book and I have some questions about Gauss sum. The Gauss sum is defined in the end of page 4, formula (1.14), by \begin{align} g(m,c)=\sum_{a \mod c} \left( \frac{a}{c} \right)_n ...
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Challenging problems in algebra (book recommendation)

Could you suggest me a book/web page where I can find challenging/hard problems in algebra (possibly with solutions) for an undergraduate student (groups, rings, fields, Galois theory)? Thanks in ...
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When does $\frac{\alpha^k-1}{\alpha-1}$ become a unit in $\mathbb{Z}[\alpha]$?

Let $\alpha$ be a complex number. For which $k\in\mathbb{Z}$ does $\frac{\alpha^k-1}{\alpha-1}$ become a unit in $\mathbb{Z}[\alpha]$? If $\alpha=\xi_m$, an $m$-th primitive root of unity, then ...
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2answers
35 views

How to go about this proof for non zero polynomials.

How do I go about proving this? Let $\mathbb{F}$ be a field and $X$ an indeterminate, and consider the polynomial ring $\mathbb{F}[X]$. Let $f(X), g(X) \in \mathbb{F}[X]$ with $f(X), g(X) \neq ...
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1answer
31 views

When does it hold that $a^{-1} \in \mathbb{Z}[a]$?

When does it hold that $a^{-1} \in \mathbb{Z}[a]$, for an algebraic number $a $? If $a$ is a root of unity of any order, done. But I know there are other examples: e.g., $2-\sqrt {3}$.
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Show that if $U$ and $V$ are ideals in ring $(P, +, \cdot)$ then also $U\cap V$ is ideal

Show that if $U$ and $V$ are ideals in ring $(P, +, \cdot)$ then also $U\cap V$ is ideal in $(P, +, \cdot)$ and $U+V= \left \{ u+v: u\in U, v\in V \right \}$ is an ideal in $(P, +, \cdot)$ Completely ...
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0answers
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Weight spaces of a irreducible representation of $\mathfrak{gl}(n, \mathbb{C})$.

Let $\mathfrak{gl}(n,\mathbb{C})$ be the general linear Lie algebra. Let $\{E_{s,t}\}_{1\leq s,t,\leq n}$ be the standard basis for it. And set its Cartan subalgebra $\mathfrak{h}$ to be ...
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If G = pqr where p,q,r are prime, and all the Sylow groups are normal, then is G is abelian? [on hold]

Let $G$ be a group with $|G| = pqr$ where $p,q$ and $r$ are distinct primes with $p<q<r$. If all the Sylow subgroups are normal, then is $G$ abelian? Thank you in advance,
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If the action is free, is it necessarily a covering space action?

Suppose a group $G$ acts simplicially on a $\Delta$-complex $X$, where "simplicially" means that each element of $G$ takes each simplex of $X$ onto another simplex by a linear homeomorphism. If the ...
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1answer
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Proving the following subgroup(verification of logic)

So I was reading the following theorem from dummit that is If $|H| = n <\infty$ then for each positive integer dividing n there is a unique subgroup of $H$ of order $a$. This subgroup is the ...
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1answer
15 views

Is the semidirect product of normal complementary subgroups a direct product.

If $G$ is a group with $K$ and $N$ as normal complementary subgroups of $G$, then we can form $G \cong K \rtimes_\varphi N$ where $\varphi:N \to Aut(K)$ is the usual conjugation. But someone told me ...
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1answer
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Correspondence between Ext group and extensions (from Weibel's book)

I am trying to understand the proof of Theorem 3.4.3 from Weibel's book Introduction to homological algebra. The statement is the following. Let $R$ be a ring. Given $R$-modules $A$ and $B$, an ...
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1answer
36 views

Zeroes of prime polynomials in the algebraic torus (A Hilbert's Nullstellensatz for Laurent polynomials?)

Let $Q\in\mathbb C[z_1,\dots,z_D]$ be a prime polynomial and let $Z(Q)$ be the algebraic hypersurface of its zeroes. Assume that $P\in\mathbb C[z_1,\dots,z_D]$ is a polynomial which has zeroes at ...
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1answer
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Finite covering by finitely generated B-algebras

While I was working on the first properties of schemes on Hartshorne's Books, I needed the following result that I couldn't prove it: Let $X=Spec(A)$ be an affine scheme. Suppose that there exist an ...
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2answers
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Find a Four-element Abelian Subgroup of $S_5$ [duplicate]

Prof. Charles Pinter's "A Book of Abstract Algebra" provides this exercise: Ch 7 (Groups of Permutations) Part B #3 - Find a four-element abelian sub-group of $S_5$. Write its table. Please ...
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1answer
36 views

How to find inverse of generator of a finite field?

I need to find the inverse of generator of finite field $\mathbb{F}_{2^4}$ with irreducible polynomial , $f(x)=x^4+x+1$ i.e. if $g=0010$ is the generator of this field then how to find $g^{-1}$?
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54 views

Presentation of a group question

So I know that given a presentation of a group $G$, one can derive from the relations of the group presentation any element in the group $G$ right. However, I do have some confusion. If we take ...
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quaternions - understanding a formula

Quaternions are new for me. I am trying to understand the following formula: What are: $\large{q^x}$ ? I don't think it is a power. $\large{q^t}$ ? just a transposition of the quaternion $q$? ...
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1answer
37 views

Is there an element in $(\mathbb{Z}/7161\mathbb{Z})^*$ that has order 30? [duplicate]

I'm guessing that there is an element in $(\mathbb{Z}/7161\mathbb{Z})^*$ that has order 30, but can someone give me an example or a proof that there indeed is one? Either one is okay, whichever is ...
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1answer
22 views

Roots of $X^{l-1}+1$ in a quadratic extension $F_q$, $q=l^2$

Consider a finite field $F_q$ where $q=l^2$ ($l$ can be of the form $p^m$). Does $F_q$ has a root of $X^{l-1}+1$? As $X^l+X = X(X^{l-1}+1)$ we can show that $X^{l-1}+1$ splits if it has a root. This ...
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2answers
19 views

What is an algorithm for describing the partition of this equivalence relation?

Let $\mathbb{R}$ be the set of real numbers,$ f : \mathbb{R} \to \mathbb{R}$ a map, and $E$ the equivalence relation on $ ℝ $ defined by $E = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid ...