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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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 $R=S$. Is ...
3
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1answer
1k views

How does Dummit and Foote's abstract algebra text compare to others?

I am looking for a good book on abstract algebra (and if possible linear algebra). Obviously as most of these texts are fairly expensive I want to know for sure which one is best for me. Could ...
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0answers
29 views

There is no irreducible polynomial of largest degree in $\mathbf{F}_q[x]$

I am asked to prove or disprove that given a finite field $\mathbf{F}_q$, the ring $\mathbf{F}_q[x]$ contains irreducible polynomials of arbitrarily large degree. I couldn't think of a reason why this ...
10
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1answer
439 views

On group automorphism of subgroup a group $G$

Let $G$ be a group and $H$ be a subgroup of $G$. When is $\rm{Aut}(H)$ a subgroup of $\rm{Aut}(G)$?
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1answer
13 views

Splitting field and automorphisms

I know that if $K$ is a field and $f\in K[x]$, then there exists a splitting field of $f$ on $K$. If one has two isomorphic fields $K_1$ and $K_2$ (say $\sigma$ an isomorphism) and $f\in K_1[x]$, ...
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2answers
40 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 ...
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votes
2answers
12 views

How to find the Permutation in S8 given as the product C= (1483)(12765)(34687)?

I don't understand how to answer this. Just by reading it off, "1 goes to 4, and 4 goes to 6, thus 1 goes to 6" but that logic doesn't match with the answer i've been giving: Answer: (127)(386)(45). ...
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1answer
29 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
19 views

normalization of a curve, simplest example

I am learning about normalization of nodal curves and I am trying to understand the simplest example: $xy=0$ As far as I understand its coordinate ring is $k[x]\oplus k[y]$ (let $k$ be an ...
-3
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1answer
15 views

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 $H$?
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2answers
43 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 ...
3
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0answers
35 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 ...
3
votes
1answer
72 views

Describe, as a direct sum of cyclic groups, given a map $\phi: \mathbb{Z}^{3} \longrightarrow \mathbb{Z}^{3}$

I'm trying to resolve the next one: Describe, as a direct sum of cyclic groups, the cokernel of the map $\phi: \mathbb{Z}^{3} \longrightarrow \mathbb{Z}^{3}$ given by left multiplication by the ...
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0answers
33 views

$\mathbb Z[r_1,r_2,…,r_n] =\mathbb Z[\frac 1m]$ [duplicate]

Question Let $r_1,r_2, ...,r_n \in \mathbb Q $. Then $$\mathbb Z[r_1,r_2,...,r_n] =\mathbb Z[\frac 1m]$$ for some integer $m$. I think m must be the least common multiple of the ...
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1answer
28 views

Is the Kähler differential of a continuous function ring trivial?

Suppose $A=C^0(\mathbb R)$ is the ring of real-valued continuous functions on $\mathbb R$. Is it true that, the Kähler differential $\Omega_{A/\mathbb R}$ trivial? In other words, suppose that ...
2
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1answer
45 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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0answers
51 views

algebras without identity [on hold]

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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0answers
28 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 ...
0
votes
1answer
25 views

Nilpotent matrix given nilpotent traces [on hold]

Let R be a 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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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
52 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 ...
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0answers
11 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 ...
5
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1answer
162 views

Given $f, g \in k[x,y]$ coprime, why can we find $u,v \in k[x,y]$ such that $uf + vg \in k[x]\setminus\{0\}$?

Let $k$ be a field. Given $f, g \in k[x,y]$ coprime, why can we find $u,v \in k[x,y]$ such that $uf + vg \in k[x]\setminus\{0\}$? I can do it for specific polynomials, but I'm struggling to ...
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1answer
30 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 ...
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1answer
35 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 ...
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0answers
37 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
21 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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1answer
22 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 ...
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0answers
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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
44 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 ...
4
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2answers
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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 ...
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1answer
40 views

How is a characteristic subgroup verified?

I know the definition of a characteristic subgroup: $\sigma (H)=H$ for all $\sigma \in \text{aut}\, G$ where $H \leq G$. But, I do not understand how $\sigma$ is defined. Surely we can map $H$ to $H' ...
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1answer
25 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, ...
2
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3answers
40 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 ...
3
votes
3answers
46 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 ...
3
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1answer
555 views

Why is such an ideal ambiguous?

Suppose I have an $R$-ideal $I$ with $$I=(1-\zeta)^n XR$$ with $R=\mathbb{Z}[\zeta]$, $\zeta$ a primitive $p$-th root, $X$ an ideal in the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\zeta + ...
4
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1answer
56 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 ...
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2answers
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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 ...
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1answer
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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 ...
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1answer
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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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3answers
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Proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain

We're proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain, using the norm function $$\nu (a + b\sqrt{2} ) = |a^2 - 2b^2|$$ and the first part says that since $\nu (a + b\sqrt{2} ) = |(a + ...
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1answer
37 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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1answer
32 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

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

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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7 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 ...
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0answers
15 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 ...
3
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3answers
447 views

Universal Mapping Property of Free Abelian Groups

Let S be a set and $F=F_S$ the free group on S. Let $F'$ be the commutator subgroup of $F$. Set $A=A_S = F/F'$, and call it the free Abelian group on $S$. Prove the universal mapping property of the ...
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2answers
46 views

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

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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+100

Show that the sequence is exact

We have that $R$ is a commutative ring. Suppose that $0\rightarrow A\rightarrow B\overset{f}{\rightarrow} C\rightarrow 0$ and $0\rightarrow C\overset{g}\rightarrow D\rightarrow E\rightarrow 0$ are ...
0
votes
1answer
39 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. ...