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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Let $H$ be all permutations that leave fixed 6, then $H$ is a subgroup of $S_7$? [on hold]

If in $S_7$ take as $H$ the set of all permutations that leave fixed to 6, it is a subgroup H? How can I get all elements of $H$?
1
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2answers
56 views

Prove that $[a]=[b]$ iff $a\sim b$.

If $[a]=[b]$ then $a$ is in $[b]$ and $b$ is in $[a]$. If $a \sim b$, then $[a]$ is a subset of $[b]$ and $[b]$ is a subset of $[a]$. Then using the equivalence properties and showing that there is a ...
4
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0answers
52 views

Shortest proof for showing $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID.

I'm looking for an easy proof for that $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID. One proof I know is to show that the field norm is a Dedekind-Hasse norm, but this proof is quite dirty( that it ...
2
votes
2answers
49 views

Irreducibility of polynomials in $\mathbf{Z}_p[x]$ - understanding proofs

I am reading through some irreducibility proofs and there's something I don't quite understand: $x^3+2x+1$ is irreducible in $\mathbf{Z}_3[x]:$ no roots in $\mathbf{Z}_3$ and degree $3$ so ...
1
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0answers
54 views

Value group of simple field extension: Are these value groups equal?

Suppose that a field extension $L/K$ is finite, K is a Henselian field with a exponential valuation $v$, and $w$ is an extension of $v$ to $L$. (If it is necessary, we can also assume that $L/K$ is ...
0
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0answers
35 views

Construction of Permutation Group (bounded order) from a Set of Permutation

Given a set of permutation $S$. It has $|S|$ (The cardinality of $S$) elements. Consider a subset $A\subset S$. We can construct a group using $A$. One possible algorithm could be- We start ...
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0answers
73 views

algebras without identity

This problem is an exercise from Drozd-Kirichenko's book Finite Dimensional Algebras, page 29. Let $k$ be a field. Let $A$ be a $k$-algebra not necessarily with identity. Let $\overline A$ be the ...
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1answer
32 views

Nilpotent matrix given nilpotent traces [on hold]

Let R be a conmutative ring and X a two by two matrix. Supose that Tr(X) and Tr(X^2) are nilpotent elements. Prove that 2X is nilpotent. Thanks a lot.
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0answers
57 views

proving that $8x^3-6x-1$ is irreducible over $\mathbb{Q}$

When considering the impossibility of trisecting the 60 degree angle, one comes across the polynomial $f(x)=8x^3-6x-1$, which I want to prove is irreducible over $\mathbb{Q}$. I reduced the ...
1
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0answers
21 views

Decomposition of semi-simple Lie algebras

Background: Let $G$ be a finite-dimensional Lie group, with Lie algebra $L(G)$. A subspace $I\subset L(G)$ which satisfies $[L(G),I]\subset I$ is called an ideal of $L(G)$. A non-abelian Lie algebra ...
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0answers
37 views

Application of generalized Chinese remainder theorem

Question Consider the ring $\mathbb Z[x]$ and define the ideals $I_p=(px-1)$ where p is prime Prove that $\mathbb Z[x]/I_2I_3...I_p$ is isomorphic to $\{\frac{n}{2^{a_2}3^{a_3}...p^{a_p}} | ...
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0answers
42 views

How many elements are in the field of fractions $\Bbb Z_3(t)$?

As in exercise for my Galois Theory course I am supposed to find the number of elements in the field of fractions $\Bbb Z_3(t)$. I am very confused as to how to approach this question because I ...
0
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1answer
31 views

Groups actions and Represenations

I'm reading Basic Algebra by Knapp and on Page 161 of Chapter 4 Section 6 he's talking about how a group action on a set can define other group actions. I can see how this makes sense in theory, but ...
1
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1answer
39 views

Galois Extension.

Suppose K is a finite field extension of $\mathbb{Q}$. Let K ⊆ L be a Galois field extension and K ⊆ K′ be a finite field extension. Show that K′ ⊆ K′L is a Galois field extension and $$\text{Gal}(K′L ...
1
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1answer
27 views

Intermediate field extensions of $\mathbb{Q}(\zeta_3)/\mathbb{Q}$

I'm trying to determine the intermediate field extensions of $\mathbb{Q}(\zeta_3)/\mathbb{Q}$. The minimal polynomial of $\zeta_3$ is $x^3+1$, which has roots $\zeta_3, \zeta_3^2$ and $-1$. Therefore, ...
4
votes
2answers
93 views

Why is $\mathbb{Q}(\sqrt{2})\subseteq\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})\subseteq\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$

In office hours yesterday my instructor said $\mathbb{Q}(\sqrt{2})\subseteq\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})\subseteq\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$. I know ...
0
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1answer
22 views

Suppose $P, N$ are sub-modules of a module $M$. Then there exist natural isomorphisms..

Suppose $P, N$ are sub-modules of a module $M$. Then there exist natural isomorphisms $$(1) \ \ \frac{P+N}{P} \approx \frac{N}{N \cap P} \ \ \text{ and } \ (2)\ \ \frac{P+N}{N} \approx ...
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2answers
23 views

How to prove the uniqueness of multiplicative identity?

Suppose $i_1, i_2 \in R$ which are multiplicative identity. Let $a$ also be in $R$. Then $a*i_1=a$ which means $a=i_2$. Thus, $i_2*i_1=i_2$. Now $a*i_2=a$, then $a=i_1$ hence $i_1*i_2=i_1$. Now how do ...
2
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1answer
29 views

How do I prove the uniqueness of additive identity?

First, suppose $i_1$ and $i_2$ are additive identity in ring R. From the definition of "additive identity" $a+i_1=a$ such that there is $a$ $\in$ $R$, including for $a=i_2$, so $i_2+i_1=i_2$. But ...
0
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0answers
57 views

Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.

I would like to have a result similar to "Traces of all positive powers of a matrix are zero implies it is nilpotent". Namely: Let $R$ be a commutative $\mathbb{C}$-algebra, $A \in ...
1
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1answer
40 views

Coprime polynomials in $k[x,y]$ are also coprime in $k(y)[x]$

Let $f,g \in k[x,y]$ be polynomials with no common factor. Prove that when viewed as elements of $k(y)[x]$ they still do not have a common factor. Say we have $f=\sum a_{ij}x^iy^j,\ g=\sum ...
4
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2answers
48 views

Integrally Closed domain and Principal Ideal

Let $R$ be an integrally closed local domain. Suppose there is a $y\in I^n$ such that $yI^n=I^{2n}$ for some $n$. I would like to prove that $I^n=(y)$. Source: The above question comes from the ...
2
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3answers
45 views

Show that $E$ is the splitting field of some polynomial in $F[x].$

Let $K$ be the splitting field of some polynomial over $F$. If $E$ is a field extension contained in $K$ and $[E:F]=2,$ I want to show that $E$ is the splitting field of some polynomial in $F[x].$ I ...
0
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1answer
38 views

Question on the Notation of an Abstract Algebra Question

The following is a question that I came across in a textbook I'm reviewing for self-study. The book is "Introduction to Abstract Algebra", 4th Edition, by W. Keith Nicholson. I have a question both ...
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0answers
8 views

$G$-invariant symmetric, nondegenerate form is unique up to scalar

Let $V$ be a f.d. representation of a finite group $G$ over a field $F$. A standard argument shows there is a $G$-invariant, symmetric, nondegenerate bilinear form on $V$. If $(-,-)$ is any such ...
2
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1answer
36 views

algebraic system of planes

I want to get a picture of intersection of several planes in 3d, with their algebraic system something like this: (Large version)
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1answer
34 views

Complex Roots of Unity?

I just had a question about complex roots of unity. It's not a computation thing; I know how to find them and I know what they mean. In my class last semester, my professor mentioned that they are ...
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0answers
18 views

What are the good references on tame hereditary algebras?

I have Thomas Brustle's Typical Examples of Tame Algebras, but I still do not have a systemic understanding of what tubes are and what regular roots of a tame hereditary algebra are. I'm also looking ...
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0answers
24 views

Has the characterization of non-unique factorizations been studied in a general context?

In this paper, a theory of principalization fields is introduced, that lets the possible factorizations of an element of an algebraic number field be characterized as groupings of the unique ...
0
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1answer
14 views

Monoid operation order sensitive?

It is a basic question, none the less I cannot find an answer: A monoid is associative (with an identity) (m1∙m2)∙m3=m1∙(m2∙m3). e∙m=m∙e=m If you consider a monoid over natural numbers (N,+,0) for ...
0
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1answer
42 views

The number of Sylow $5$-subgroups in $S_6$

Find the number of sylow $5$-subgroups in $S_6$. First: $ord(S_6)=6!=2^4\cdot 3^2\cdot 5=144\cdot 5$, so $n_5|144$ and $n_5\equiv 1\pmod5$, where $n_5$ is the number of sylow $5$-subgroups. ...
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0answers
42 views

Conjugation in algebraic number theory

Let $K$ be an algebraic number field of deg $n$ over $\mathbb Q$, then given $\alpha \in$ $O_k$ its ring of integers, we can choose a $\mathbb Q$-basis $\omega_1, \omega_2, ...,\omega_n$ of $K$ s.t. ...
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1answer
37 views

Prove that in a Euclidean domain nonzero prime ideals are maximal. [closed]

It is from masters qualifying exam. I am an undergraduate student. I want to wonder this proof. Can you prove to explain clearly?
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2answers
47 views

A group of order pq with a single subgroup of order p [closed]

Given a group $G$ of order $pq$ (such that $p,q$ are primes and $p < q$) that have a single subgroup of order p (named $H$) prove that $\forall h \in H , g\in G : ghg^{-1} = h$
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1answer
47 views

Prove that if $xy=yz$ then $ \exists u,v \in A^*$ and $\exists p \in\mathbb N$ such that $x=uv$, $z=vu$ and $y=(uv)^pu$

Prove that if $xy=yz$ then $ \exists u,v \in A^*$ and $\exists p \in \mathbb{N} $ such that $x=uv$, $z=vu$ and $y=(uv)^pu$. $A^*$ is the set of all words that can be formed over the alphabet $A$. By ...
0
votes
1answer
15 views

Product of permutations and subgroup generated by permutation(s)

I'm get confused while working with permutations, so I have some questions. $\sigma$ = (1,7,3)(2,10,4,8) $\rho$ = (3,7) $\tau$ = (1,7) First I am told to compute $\tau$$\sigma$$\tau^{-1}$ I dont ...
0
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2answers
36 views

How many subgroups of order 25 in $G = \Bbb{Z}_{360} \oplus \Bbb{Z}_{150} \oplus \Bbb{Z}_{75} \oplus \Bbb{Z}_{3}$

Let $G = \Bbb{Z}_{360} \oplus \Bbb{Z}_{150} \oplus \Bbb{Z}_{75} \oplus \Bbb{Z}_{3}$ a. How many elments of order 5 in $G$ b. How many elments of order 25 in $G$ c. How many elments of order 35 in ...
0
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1answer
24 views

Write $A=Aut(C_{7^3})$ as a direct product of cyclic groups

Write $A=Aut(C_{7^3})$ as a direct product of cyclic groups of a prime order $$A \cong C_{{p_1}^{m_1}} \times \ldots C_{{p_n}^{m_n}}$$ where p are prime numbers there is a theorem that if ...
2
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2answers
40 views

on finite division subring of a ring

Is there any example of a ring which is not a division ring but any of its subring is a division ring? According to me if $R$ is a ring and $S$ is a division subring then $1\in S$ and hence ...
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31 views

Hall $\pi$-subgroup of a minimal normal subgroup of $G$

Let $B$ be a minimal normal subgroup of $G$ and suppose that $H \in$ Hall$_\pi$($B$). Then $B = S_1 \times \dots \times S_n$ where $S_i$ are simple groups. I'm not sure how $H = \langle H \cap S_i ...
2
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1answer
56 views

Is the primitive element of an extension of a finite field expressible as a linear combination of adjoined elements?

Suppose $F$ is a field, and $\alpha,\beta$ are algebraic, separable elements over $F$. If $F$ is infinite, the proof of the primitive element theorem gives some $c\in F$ such that ...
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33 views

Factorization of Polynomials over a Field: Question on Theorem

In Fraleigh's Abstract Algebra book this corollary is written and proven shortly after Gauss's lemma is introduced: Corollary: If $f(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_0$ is in $\mathbb{Z}[X]$ ...
1
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1answer
35 views

Even functions absorb composition?

If $f(x)$ and $g(x)$ are real functions and $g$ is even, so is $f(g(x))$. Even functions are also closed under addition. I noticed that these are similar properties to those of an ideal of a ring, ...
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1answer
30 views

Does $K/E$ and $E/F$ being normal mean $K/F$ is normal?

Let $F\subset E \subset K$ be fields. Suppose that $K/E$ and $E/F$ are normal. Is $K/F$ also normal? I feel that this statement is not true in general but I cannot find a counter-example. Any ...
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0answers
10 views

superset of commutator subgroup

If $N$ is a commutator subgroup of $G$ and $H$ is a subgroup of $G$ with $N\subseteq H$, show that $H\triangleleft G$! I've shown that $N\triangleleft G$ and $G/N$ is abelian but i don't have idea to ...
2
votes
1answer
45 views

Properties of the norm in a Euclidean Domain

I am aware of the fact that the Euclidean Norm does not need to be unique in a given domain, however my question is essentially: can we ensure that the properties of the norm remain the same? More ...
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0answers
23 views

Show that exist unique $m_{f}$ such that $f(a+n\mathbb{Z})=m_{f}a+n\mathbb{Z}$ and $f$ is an automorphism

Let $n\in\mathbb{N}$ and $f:(\mathbb{Z}/n\mathbb{Z},+)\to (\mathbb{Z}/n\mathbb{Z},+)$ is a homomorphis of group. Show that exist unique $m_{f}\in\{0,...,n-1\}$ such that ...
0
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0answers
38 views

Non-existence of a particular type of tower of number fields

I have number fields $\mathbb{Q}\subset K\subset H$ where $K\subset H$ is Galois. I want to show that is is impossible for a rational prime $p\in\mathbb{Z}$ to remain first inert in $K$ but then for ...
3
votes
3answers
47 views

What polynomial maps to $i$ under $\mathbb{Q}[x] \to \mathbb{Q}[x]/(x^2+1) \simeq \mathbb{Q}[i]$?

The rings $\mathbb{Q}[i]$ and $\mathbb{Q}[x]/(x^2+1)$ are isomorphic, and there is a surjective ring homomorphism from $\mathbb{Q}[x]$ to $\mathbb{Q}[x]/(x^2+1)$. Can someone give me an example of ...
1
vote
2answers
54 views

Find $Gal(K/\mathbb{Q})$ and show that $K/\mathbb{Q}$ is normal where $K=\mathbb{Q}(a)$

Let $K=\mathbb{Q}(a)$ and $a$ is a root of $x^3+x^2-2x-1 \in \mathbb{Q}[x]$. Find $Gal(K/\mathbb{Q})$ and prove that $K/\mathbb{Q}$ is normal. I just noticed that $a^2-2$ is also a root of the ...