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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3
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2answers
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All ideals of a subring of $\Bbb Q$

Let $p$ be prime and $R=\{\frac{a}{b}:a,b \in \Bbb Z,b \neq0\text{ and }p\nmid b\}$. As an exercise, I have to prove that $R$ is a subring of $\Bbb Q$. My idea: With $a = 1$ and $b = 1$, $\frac{...
-1
votes
1answer
36 views

Are there two groups $G_1 , G_2 $ of orders $7$ and $8$ respectively & a morphism $f: G_1 \to G_2 $ such that $|\operatorname{Im}(f)| = 4$?

Is is possible to find two groups $G_1, G_2 $ of orders $7$ and $8$ respectively & a morphism $f: G_1 \to G_2$ such that $|\operatorname{Im}(f)| = 4$ ?
0
votes
1answer
33 views

Prove that this set is a group with usual multiplication [closed]

Prove that the set of all numbers of the form: $\ P1 + P2*{\sqrt d} $ , (P1 & P2 are elements of All Quotient Numbers), (P1^2 + P^2>0) & (d is an element of All Complex Numbers) not being a ...
1
vote
1answer
40 views

Let $p$ be prime. If an INFINITE group has more than $p-1$ elements of order $p$, why can't the group be cyclic?

Let $p$ be prime. If a group has more than $p-1$ elements of order $p$, why can't the group be cyclic? I understand how to prove this if the group is finite because the contrapositive of this ...
0
votes
2answers
60 views

Ideals and Tensor Products

I'm reading Osbourne's Basic Homological Algebra, and on page 18 he has this situation where we've got a ring $R$ and a right-ideal $I$, and some left $R$-module $B$. He says $I\otimes B$ is not a ...
1
vote
0answers
29 views

If the group is infinite , what inference should I make about the number of nonidentity elements that satisfy the equation $x^5=e$?

In a finite group, show that the number of nonidentity elements that satisfy the equation $x^5=e$ is a multiple of 4. If the stipulation that the group be finite is omitted, what can you say about the ...
2
votes
0answers
48 views

When is a finite $R$-algebra isomorphic to $R$?

Let $R$ be a $\bar{k}$-algebra (of finite type or complete) reduced (and maybe integral, if needed), let $A$ be an $R$-algebra, finite as an $R$-module, reduced and connected and such that there ...
2
votes
0answers
21 views

Notation for subring of quotient ring

Let $S$ be a subring and $I$ an ideal of the ring $R$. Is there some standard notation for the subring of $R/I$ given by $\{ s + I: s \in S \}$. Is it appropriate to write $S/I$ even though $I$ is not ...
0
votes
1answer
27 views

Permutation Product

Here is a problem from Algebra by Michael Artin: $p = \left( {\begin{array}{*{20}{c}} 3&4&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&5 \end{array}} \right),q = \left( {\...
4
votes
0answers
96 views

Is $\pi^\pi$ algebraic over $\mathbb{Q}(\pi)$?

Is $\pi^\pi$ algebraic over $\mathbb{Q}(\pi)$? I have a feeling that it's a rather easy question, but since my understanding of field extensions is only superficial I really can't handle this ...
0
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1answer
45 views

Properties of Short Exact Sequences

Some of the work I have been doing lately is heavily dependent on chasing commutative diagrams so I have been brushing up on short exact sequences since I was not familiar with them. For the most part ...
3
votes
0answers
37 views

$\mathbb{T}\text{-Alg(Set)}$ is complete and cocomplete.

Let $\mathbb{T}$ be a finitary algebraic theory and $\mathbb{T}\text{-Alg(Set)}$ be the category of finite-product-preserving functors $\mathbb{T} \rightarrow \text{Set}$. It is written in my ...
0
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0answers
53 views
+100

Growth of the characters of finite permutation groups in the number of symbols

I have the following questions. When can the characters of the irreducible representations of the elements of a finite permutation group increase exponentially in the number of the symbols the ...
2
votes
1answer
26 views

Is the tensor algebra functor a strong monoidal functor?

Let $K$ be a commutative ring. Is it true that $T(V \oplus W) \cong T(V) \otimes_K T(W)$ as $K$-algebras for any $K$-modules $V$ and $W$? The reason I ask is that I have heard it mentioned (I don't ...
1
vote
1answer
56 views

Can multiple Boolean variables and equations be converted to a single integer variable and multiple modulo equations?

e.g. let $x,y,z \in \mathbb{B}$ (Boolean) and $w \in \mathbb{Z}$ (integers) and $p,q,r \in \mathbb{P}$ (primes) For $x$ let $(0,1)$ be represented by integers $(\overline{a},a)$ mod p For $y$ let $...
0
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0answers
17 views

When does the unit of universal enveloping algebra adjunction fail to be injective?

The Poincaré–Birkhoff-Witt (PBW) theorem implies that if $K$ is a commutative ring and the Lie $K$-algebra $\mathfrak{g}$ is a free $K$-module, then the unit $\eta_{\mathfrak{g}}: \mathfrak{g} \to U (\...
35
votes
11answers
2k views

Amazing isomorphisms [closed]

Just as a recreational topic, what group/ring/other algebraic structure isomorphisms you know that seem unusual, or downright unintuitive? Are there such structures which we don't yet know whether ...
0
votes
1answer
50 views

why the algebra $A = k[x]/(x^n)$ has finite representation type? [closed]

Suppose that $k$ is algebraically closed. Then why the algebra $A = k[x]/(x^n)$ has finite representation type? please clarify the answer.
0
votes
0answers
55 views

Is $\pi e$ irrational? [duplicate]

During our ongoing research, we need to prove that $\pi e<\lceil \pi e\rceil$. Is $\pi e$ irrational? How to prove it? Thanks- mike
-1
votes
0answers
66 views

Isomorphism between a quotient of a polynomial ring and a polynomial ring [on hold]

I'm going to show that $\mathbb{K}[x,y,z]/(y^2-xz)$ is not isomorphic to any polynomial ring. I'll be grateful of someone brings a hint to show this result. Thank you.
0
votes
1answer
24 views

Question about $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$.

I am confused about the set $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$... Could someone please explain me why this corresponds to the set of pairs of symmetric $3$ by $3$ matrices? Thank you!
2
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0answers
39 views

To prove: The intersection of all normal subgroups of finite index of a free group is trivial.

Let $S$ be a set, $G$ its free group and $\mathcal{N}$ the set of normal $N \leq G$ such that $[G:N] \leq \infty$. Prove that $$ \bigcap_{N \in \mathcal{N}} N \ = \ \{ e\} $$ I know how free ...
-5
votes
1answer
34 views

In $\mathbb{Z}[x,y]$, $\langle x+y,x-y \rangle \subsetneqq \langle x,y \rangle$ [closed]

In $(\mathbb{Z}[x,y],+,\cdot)$ show that $\langle x+y,x-y \rangle \subsetneqq \langle x,y \rangle$. In $(\mathbb{Q}[x,y],+,\cdot)$ show that $\langle x^2+y^2+2xy \rangle \subsetneqq \langle x^2+y^...
1
vote
1answer
34 views

Why is SU(2) not the same group as T3?

Why is $S^3 = SU(2)$ not the same group as $ S^1 \times S^1 \times S^1 $? It seems that, the $T^3$ torus is abelian, wheras $S^3$ is not. Is that enough? Problem 2.6.5 from John Stillwell, Naive ...
0
votes
1answer
55 views

About the definition of polynomials on vector spaces?

In the book Linear Systems Theory and Introductory Algebraic Geometry (R. Hermann) the author defines A polinomial on V (a $\mathbb K$-vector space) is an element of the smallest ...
1
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0answers
47 views

Clifford algebra of a non-diagonal quadratic form over rings

I know how to construct explicitly the clifford algebra of a quadratic form over fields, even in the case the diagonal quadratic form over rings. But how should I $\textbf{construct explicitly}$ the ...
-1
votes
1answer
23 views

Cycles, Abstract algebra

I'm new to abstract algebra. Question is: " What are all the elements of the cycle <(1 2)>? " Is this in fact a "cycle"? I understood that a cycle was displayed as, say, <4>, i.e. only one ...
2
votes
2answers
40 views

Let $|G|$ be an abelian group and let $H = \{g \in G : |g| \text{ divides } 12 \}.$ Prove that $H$ is a subgroup of $G$.

Let $|G|$ be an abelian group and let $H = \{g \in G : |g| \text{ divides } 12 \}.$ Prove that $H$ is a subgroup of $G$. I know that I have to show that $a,b \in H \Rightarrow ab^{-1} \in H$ or $(ab ...
1
vote
1answer
42 views

Group of order $pq$ with $p<q$ prime has $q-1$ elements of order $q$

If $p<q$ are primes and $G$ is a group of order $pq$, then $G$ has exactly $q-1$ elements of order $q$. My attempt was: Use Sylow theorem to show that $G$ has only one subgroup of order $q$, so ...
0
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1answer
11 views

How the symmetryis related to lie groups? what are the applications? [closed]

lie groups and it's applications are required.Also how symmetrical examples fitted in that?
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0answers
27 views

must this extension of a DVR be unramified?

Let $A$ be a normal domain, and $P$ a height 1 prime, then $A_P$ is a DVR. Let $K$ be the fraction field of $A_P$, and let $L$ be a finite Galois extension of $K$ of degree $e$, let $B$ be the ...
1
vote
4answers
64 views

Showing that two quotient rings are isomorphic

Is $\mathbb{Q}[x]/(x^2-x-1)$ isomorphic to $\mathbb{Q}[x]/(x^2-5)?$ My guess is yes. I am trying to find an isomorphism between the two. Universal Property of Quotient certainly helps. I am thinking ...
2
votes
1answer
63 views

What is $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p(X)} \mathbb{F}_p(\sqrt{X})$?

I am trying to understand what $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p (X)} \mathbb{F}_p(\sqrt{X})$ is. $\mathbb{F}_p(\sqrt{X})$ is the field of rational functions in $\sqrt{X}$. What is it ...
10
votes
2answers
487 views

How to understand “tensor” in commutative algebra?

Tensor is sure an important concept in commutative algebra, but the definition is kind of abstract, so is there any way to understand it which is easier? Thanks advance! The definition I see is the ...
0
votes
0answers
32 views

does algebra over algebraically closed field has isomorphism classes of irreducible modules?

Let $F$ be an algebraically closed field and $M$ be an F-algebra. Which are the conditions that make $M$ has finitely many isomorphism classes of irreducible $M$-modules?
1
vote
1answer
33 views

Counting maximal subgroups in a finite $p$-group

Let $G$ be a finite $p$-group. I want to show that if the number of maximal subgroups is strictly less than $p+1$ then $G$ is cyclic. This may not be true, but if the number of maximal subgroups is ...
0
votes
0answers
25 views

Show that $\mathbb{C}\otimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\times\mathbb{C}$ [duplicate]

Show that $\mathbb{Q}(\sqrt{2})\otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{3})\simeq\mathbb{Q}(\sqrt{2},\sqrt{3})$ and $\mathbb{C}\otimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\times\mathbb{C}$ I approached ...
2
votes
2answers
37 views

Is there an infinite field F with char(F)=p and not algebraically closed field?

Is there an infinite field F with characteristic of the field $F$ is $p$ (p is prime) and not algebraically closed field ?
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0answers
35 views

Sifting algorithm for group generated by a set. [closed]

On page 38 of "Lecture Notes in Computer Science" by Christoph M. Hoffmann, there is an algorithm (ALGORITHM 2). I have some confusions. The algorithm needs to go to all column element indexed by ...
4
votes
2answers
73 views

$\mathbb Z$ basis of the module $\mathbb Z [\zeta]$

Given an $n$-th root of unity $\zeta$, consider the $\mathbb Z$-module $M := \mathbb Z[\zeta]$. Does this module have a special name? Does a basis exist for every $n$? And if so, is there an ...
2
votes
2answers
35 views

Embeddings of pure cubic field in complex field

I know that the complex embeddings (purely real included) for quadratic field $\mathbb{Q}[\sqrt{m}]$ where $m$ is square free integer, are $a+b\sqrt{m} \mapsto a+b\sqrt{m}$ $a+b\sqrt{m} \mapsto a-b\...
3
votes
3answers
55 views

$|G| = ?$ if its subgroups are $\{e\}$, $G$ itself, and a subgroup of order $7$?

Suppose that a cyclic group $G$ has exactly three subgroups: $G$ itself, $\{ e \}$, and a subgroup of order $7$. What is $|G|$? What can you say if $7$ is replaced with $p$ where $p$ is prime? I ...
4
votes
1answer
50 views

Is $x^{2\cdot 3^n}+x^{3^n}+1$ irreducible (mod 2)?

I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, it seems that the polynomials of the form $$x^{2\cdot3^n}+x^{3^n}+1 \pmod 2$$ are irreducible. ...
0
votes
0answers
20 views

Affine and linear reflections

Let $\gamma$ - affine reflection in complex space, which is transformation with properties: (1) $\gamma$ is a motion (thus linear part of $\gamma$ : $\mathbf{Lin} \gamma \in U(V)$), (2) $\gamma$ ...
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0answers
35 views

Tensor product for vectors

This might be a simple question but I'm struggling to understand the tensor product of two vectors. From what I understand if $\vec{v}$ and $\vec{w}$ have dimensions $2$ and $3$ respectively. Then the ...
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0answers
22 views

To prove: a product of profinite groups is again profinite.

I got stuck on this syllabus about abstract algebra: Let $J$ be a set and $\{ \pi_J \ : \ j \in J\}$ be a collection of profinite groups. Show that $\prod_{j \in J} \pi_j$ is again profinite. ...
11
votes
0answers
66 views

Affine group, $[L : \mathbb{Q}] = n\varphi(n)$ or ${1\over2}n\varphi(n)$

Let $L$ be the Galois closure of $K = \mathbb{Q}(\sqrt[n]{a})$, where $a \in \mathbb{Q}$, $a > 0$ and suppose $[K : \mathbb{Q}] = n$. How do I see that $[L: \mathbb{Q}] = n\varphi(n)$ or ${1\over2}...
3
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0answers
43 views
+100

Does every finite dimensional real nil algebra admit a multiplicative basis?

We say that a finite dimensional real commutative and associative algebra $\mathcal{A}$ is nil if every element $a \in \mathcal{A}$ is nilpotent. By multiplicative basis, I mean a basis $\{ v_1, \...
4
votes
0answers
47 views

On the relationship between $\text{SL}_2(5)$ and $A_5$ [duplicate]

I have two questions. What is the quickest way to see from scratch that $\text{SL}_2(5)/\{\pm I\}$ is isomorphic to the alternating group $A_5$? Does $\text{SL}_2(5)$ have any subgroups isomorphic ...
7
votes
1answer
56 views

If $B$ is an ideal of $A$ then $B[x]$ is an ideal of $A[x]$ - what's wrong with my proof?

This is exercise E.2 from chapter 24 of Pinter's A Book of Abstract Algebra: If $B$ is an ideal of $A$, $B[x]$ is not necessarily an ideal of $A[x]$. Give an example to prove this contention. It ...