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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2answers
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Number of irreducible polynomial over a field. [closed]

Find the number of irreducible monic polynomials of degree $2$ over a field with five elements. Please anyone help me.
2
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2answers
59 views

Showing $\mathbb Z[x]/<5,x^3+x+1>$ is a field

I want to show that $ \mathbb Z[x] /<5,x^3+x+1>$ I am currently self-studying and it is a problem of previous graduate school entrance exam. I studied abstract algebra through Fraleigh's book, ...
0
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1answer
17 views

Problem in solving a question related to symmetric group. [closed]

The question is : For $n \geq 4$ prove that $S_n$ the symmetric group is generated by $n-1$ elements of order $2$. How can I solve it?Please help me.Thank you in advance.
3
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1answer
65 views

How does Gauss's lemma follow from Nagata's lemma?

In section 4 of Samuel's Unique Factorization it's said Gauss' lemma is an easy consequence of Nagata's lemma. How does this work, i.e., how to deduce Gauss' lemma from Nagata's lemma? I'm asking ...
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1answer
17 views

Field structure of non solvable field extensions

I was considering the base field $D$ which is some solvable extenstion of $ \mathbb{Q}$, and a polynomial that isn't solvable in radicals such as $ x^5 - x + 1$. If we let $\zeta$ be a root of this ...
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0answers
17 views

field trace function is linear

Assume $E / F$ is a finite extension, the trace function is defined as $$\operatorname{Tr}(a_1)=[E:F(a_1)](a_1+a_2+\ldots+a_n)$$ (where $a_j$ are all of the roots of $\min(a_1,F)$). Then, what I want ...
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0answers
46 views

a simple problem about commutative algebra [duplicate]

I think this problem seem easy, but I have no idea to approach this. Let $R$ is commutative Ring, a non-zero $f\in R[X]$ is zerodivisor if and only if there exists no-zero $c\in R$ such that $c.f=0$...
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32 views
1
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1answer
49 views

Example of a non-Kummer totally tamely ramified Galois extension

Let $A$ be a DVR with fraction field $K$, and let $L$ be a totally tamely ramified finite Galois extension of $K$ of degree $e$ - ie, the integral closure $B$ of $A$ in $L$ is a DVR with ramification ...
2
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3answers
91 views

Let $a$ and $b$ belong to a group. If $|a|$ and $|b|$ are relatively prime, show that $\langle a \rangle \cap \langle b \rangle = \{e \}$ [duplicate]

Let $a$ and $b$ belong to a group. If $|a|$ and $|b|$ are relatively prime, show that $\langle a \rangle \cap \langle b \rangle = \{e \}$. The most I can figure to do is to let $a^k = b^j$ for some $...
0
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2answers
30 views

Listing elements of subgroup generated by $\{12,42\}$ in the integers with addition

The subgroup generated by these elements should contain both $12\mathbb{Z}$ and $42\mathbb{Z}$ but also ideals of the form $$ (12k+42j)\mathbb{Z},\;j,k\in\mathbb{Z} $$ Is this the best answer I can ...
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0answers
35 views

branched cover of projective line over rationals

I was reading something when I came across the following phrase "branched cover of projective line over rational " . To understand what does author mean , I started reading , Now I know about ...
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0answers
26 views

$\Bbb Z$-graded ring with no nonzero homogeneous prime ideals

Exercise $2.18$ in Eisenbud's algebra book asks to prove: Suppose $R=\bigoplus_{n=-\infty}^\infty R_n$ is a $\Bbb Z$-graded ring such that any homogeneous prime ideal is zero. Prove $R_0$ is a field. ...
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1answer
26 views

Extensions between integral domains give extensions of fields of the same degree.

Assume that $S \subset R$ is a ring extension where, both $S$, $R$ are integral domains. Furthermore, assume that $R$ is a free $S$-module of rank $n$. Is it true that the extension of fields $\mathrm{...
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0answers
22 views

How to make check matrix H when you have generator matrix (algorithm)

It's all built on top of python numpy lib. So we have a class finite field and get access to elements of field like Finite_field[index_of_element]. Elements of field are numpy matrices(ndarray). For a ...
3
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2answers
82 views

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{...
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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 $|\operatorname{Im}(f)| = 4$ ?
1
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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
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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 ...
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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
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0answers
50 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
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0answers
22 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
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1answer
28 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( {\...
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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
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0answers
39 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 ...
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0answers
64 views

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
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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 ...
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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 $...
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0answers
20 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
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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
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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.
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0answers
56 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
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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
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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 ...
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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 ...
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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 ...
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0answers
53 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 ...
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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 ...
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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 ...
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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 ...
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0answers
29 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 ...
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4answers
66 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
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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 ...
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2answers
492 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 ...
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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?
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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
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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
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2answers
41 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 ?