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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15 views

How to make $S\otimes_R M$ into a left $S[G]$-module

I have a basic question on tensor products which might have been asked before-sorry if this is the case. Let $G$ be a finite group and $R\subset S$ any (unital) ring extension. Let $M$ be a finitely ...
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4answers
44 views

Why is the 'Law of Cancellation' for groups only an implication?

It is easy to see that for a Group $G$ and $a,b \in G$ $ab = ac \Rightarrow b = c$ (See also here) But what is about the other direction? That is: $b = c \Rightarrow ab = ac$ Does this ...
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1answer
37 views

Tensor product (confusing question)

Let's say I have a tensor product $A\otimes B$ of algebras $A,B$. I have a linearly independent subset $S\subseteq A\otimes B$ such that $span(S)\cong C$, where $C$ is another algebra. Similarly, ...
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0answers
13 views

Find the number of ways of coloring pentagonal faces of the dihedral in three colors. [on hold]

My student from junior high asked me this question. I have been surfing all over the place. And I know that i should use Burnside lemma. I learnt something about it here ...
0
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1answer
35 views

When is tensor product isomorphic to product?

Let $A$ and $B$ be algebras. When do we have $A\otimes B\cong AB$, where $$AB=\{\sum a_ib_i\mid a_i\in A, b_i\in B\}$$ Is commutativity $ab=ba$ for $a\in A$, $b\in B$ a sufficient condition? Thanks ...
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2answers
25 views

Nilpotent elements of polynomial quotient ring [on hold]

Let $F$ be a field and let $f\in F[x]$ be an irreducible polynomial. Are the non-units of $F[x]/(f^n)$ nilpotent elements?
2
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1answer
47 views

Inner Product of center Z(G) of a Group

Let $G$ be a group and $Z(G)$ be its center. For $n\in \mathbb{N}$, define $$J_n=\{(g_1,g_2,...,g_n)\in Z(G)\times Z(G)\times\cdots\times Z(G): g_1g_2\cdots g_n=e\}.$$ As a subset of the direct ...
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1answer
64 views

What is the name of people who do algebra? [on hold]

People who do topology is called topologists, people who do analysis is called analysts, people who do geometry is called geometers, then how about algebra?
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0answers
24 views

If $E$ is finite, show that $E=F(u)$ for some $u\in E$.

Let $E\supseteq F$ be fields. If $E$ is finite, show that $E=F(u)$ for some $u\in E$. What I've come up with so far: Let $u\in E$ be the primitive element of $E$. Then $E^*=\langle u \rangle$. So ...
3
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2answers
40 views

Property of group of $k \times k$ orthogonal matrices

Does the group of $k \times k$ orthogonal matrices lie on an $(k^2 - 1)$-sphere? If so, of what radius? If not, does it lie on some sort of other object?
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0answers
23 views

Cardinality of stabilizer in $PSL(2,\mathbb{Z}_7)$

Let $v$ be a non-zero vector in $\mathbb{Z}_7^2$ (up to scaling), and let P be the stabilizer of $v$ (up to scaling) in the Projective special linear group $PSL(2,\mathbb{Z}_7)$. Find the Cardinality ...
1
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1answer
63 views

Factor $x^6+x^5+x^4+x^3+x^2+x+1$ in $\mathbb{F}_2[x]$ [duplicate]

I'm trying to factor $x^6+x^5+x^4+x^3+x^2+x+1$ in $\mathbb{F}_2[x]$. But I don't know how to do that. Anyone can tell whether there is a nice way to solve all these kinds of problems?
0
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0answers
18 views

$\mathbb{Z}[\frac{1-\sqrt{-19}}{2}]$ - principal ring, but not an euclidean ring [duplicate]

I am stucked on the problem. Is there someone who could tell me why $\mathbb{Z}[\frac{1-\sqrt{-19}}{2}]$ is a principal ring, but it is not an euclidean ring?
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0answers
22 views

Proving uniqueness of an isomorphism.

Theorem: Let $\sigma:F\rightarrow\overline{F}$ be a field isomorphism. Given a monic irreducible polynomial $p$ in $F[x]$, let $u$ be a root of $p$ in an extension field $E\supseteq F$ and let $v$ be ...
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1answer
34 views

Every proper maximal subgroup of a $p$-group $P$ is normal and has index $p$.

Every proper maximal subgroup of a $p$-group $P$ is normal and has index $p$. I tried to search online by I can't get a complete proof. Take $M$ to be maximal and $Z$ to be central subgroup of order ...
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1answer
31 views

Factoring polynomials in $\Bbb Z_n$

a). Factor $f(x) = x^3 + 4x^2 + 5x + 2$ completely over $\Bbb Z_7$. b). Give two different factorizations of $x^2 + x + 8$ in $\Bbb Z_{10}[x]$. I have found the zeros of both of these but I am ...
0
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1answer
56 views

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$. I've tried factoring this into multiple forms but I can't seem to find an easy way to find the $x'$s for $x^3 + 3x + 5 = 0$. Any hints or ...
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0answers
23 views

Find the splitting field of $f=x^4+x^3+2x^2+x+1$ over $\mathbb{Q}$

Find the splitting field of $f=x^4+x^3+2x^2+x+1$ over $\mathbb{Q}$ So according to wolfram alpha, the roots of this polynomial ate $\pm i$, $-\sqrt[3]{-1}$, and $(-1)^{2/3}$. So just knowing that ...
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1answer
21 views

Two questions regarding polynomial rings.

Give an example of a natural number $n > 1$ and a polynomial $f(x) ∈ \Bbb Z_n[x]$ of degree $> 0$ that is a unit in $\Bbb Z_n[x]$. For this is set $n=2$. So then $f(x) = x \in \Bbb Z_2[x] $. ...
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0answers
21 views

Does a place $v$ of a number field $K$ ramify in $L/K$ iff $v\mid d_L$?

Let $L/K$ be an extension of number fields and $v$ be a prime (an equivalence class of valuations) of $K$ and $d_L$ the absolute discriminant of $L$. I know that a rational prime $p$ in $\mathbb Q$ ...
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1answer
23 views

How to compute $[GF(128):GF(16)]$ and $[GF(3^6):GF(3^3)]$

I'm trying to compute $[GF(128):GF(16)]$ and $[GF(3^6):GF(3^3)]$, where GF stands for Galois field. I know $128=2^7, 16=2^4,$ but what can I do next? I am really confused.
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1answer
73 views

Finding the kernel of an epimorphism onto $S_3$?

Let $\Lambda$ denote the group with presentation $\langle a,b \mid abab^{-1}a^{-1}b^{-1}\rangle$. We define the following epimorphism from $\Lambda$ onto $S_3$: $\theta: \Lambda(a,b) ...
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0answers
19 views

Partitioning the set of mappings.

The following is first two steps of an algorithm given from a research paper. I understood the first step. But please explain the second step: what does mean " Rearrange the partition according to ...
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1answer
62 views

How can I prove that $S_n$ is cyclic for all $n$? [duplicate]

I know that $S_n$ is a permutation group. $S_n$ is cyclic if there exists an element $x$ such that $\left<x\right>=S_n$. But when $n=3$, $S_n$ is not abelian then $S_n$ is not cyclic. Is my ...
-3
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1answer
51 views

Isomorphic or not: two infinite groups [on hold]

The groups $(\mathbb{C}\setminus \left \{ 0 \right \},\cdot )$ and $(\mathbb{R},+)$ are not isomorphic. So I was not clear whether this statement is true or not.
3
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0answers
37 views

Kähler differentials in an inseparable field extension

Let $L/K$ be a finite (or, more generally, algebraic) field extension. It is easy to show that if $L/K$ is separable then the $L$-vector space $\Omega_{L/K}$ of relative Kähler differentials is zero. ...
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2answers
46 views

General questions about Polynomial Rings [on hold]

I'm learning about polynomial rings in my class. My instructor and book are both spectacularly unhelpful and didn't even bother to define most of the terms in my homework. So I have some general ...
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1answer
20 views

Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its decomposition field. Show that $[F:K]$ divides $n!$.

Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its decomposition field. Show that $[F:K]$ divides $n!$. Here $[F:K]$ denotes the dimention of $F$ over $K$ as a ...
1
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1answer
25 views

$f(x) \in\Bbb Q[x]$. Prove that if $f(a + b\sqrt c) = 0$, where $a, b \in\Bbb Q$ and $\sqrt c \in\Bbb Q$ then $f(a − b\sqrt c) = 0$. [on hold]

Let $f(x)\in\Bbb Q[x]$. Prove that if $f(a + b\sqrt c) = 0$, where $a, b \in\Bbb Q$ and $\sqrt c \not\in \Bbb Q$ then $f(a − b\sqrt c) = 0$. I don't really have any idea of where to start on this. ...
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1answer
36 views

$\{(1), (12)(34), (13)(24), (14)(23)\}$ is the only non-cyclic proper subgroup of $A_4$?

What can I use to display the following: $\{(1), (12)(34), (13)(24), (14)(23)\}$ is the only non-cyclic proper subgroup of $A_4$. What I've started to do: list all the elements of $A_4$ and finding ...
4
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1answer
60 views

Prove that if Aut($G$) is the trivial group, then so is $G$? [duplicate]

Let $G$ be a finitely generated group. Show that if Aut($G$) is the trivial group, then so is $G$. I know that if Aut($G$) is the trivial group then $G$ must be abelian but I'm not sure how to ...
1
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1answer
23 views

Let R* be the set of units of R and S* be the set of units of S. Prove that f(R*) = S*.

Let R and S be commutative rings with unity $1_R$ and $1_S$ respectively, and let $f: R\to S$ be a ring isomorphism. I am at a loss. Any help is much appreciated.
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1answer
68 views

What is an order of an element of a partition"?

I'm reading a paper, in which the set of all 3^3 mappings from {0,1,2} to itself (for instance {001,020,110,121,122}, {002,010,112,011}, {0,1,2}, ...) is partitioned, after which is written two ...
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0answers
21 views

Subgroup $K$ of $G$ and $a \in G$ $\implies$ $Ka=\{ka: k \in K\}$? [on hold]

I'm asked to prove the following: Let $K$ be subgroup of group $G$ and $a \in G$. Show that $Ka=\{ka: k \in K\}$. But isn't this the definition of a right coset? What do I need to prove?
4
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1answer
37 views

Constructing well-defined epimorphism onto $S_3$?

Let $\Lambda$ denote the group with presentation $\langle a,b \mid abab^{-1}a^{-1}b^{-1}\rangle$. Construct an epimorphism from $\Lambda$ onto $S_3$, making sure to check that the function is ...
3
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3answers
57 views

Show that the extension $F \vert \mathbb{Q}$ is Galois

Let $E \subset \mathbb{R}$ a field containing $\mathbb{Q}$ such that the extension $E \vert \mathbb{Q}$ is Galois, and let $F := E(\sqrt{-1})$. Show that the extension $F \vert \mathbb{Q}$ is ...
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0answers
17 views

$U$ subset of $S$, $G$ acts transitively on $S$, show that the subsets $gU$ cover $S$ evenly

I have a finite set $S$ on which a group $G$ acts transitively. Now, I let $U$ be a subset of $S$. I want to show that the subsets $gU$ cover $S$ evenly, meaning every element of $S$ is in the same ...
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0answers
41 views

Prove that $\sin ^{-1} 1 $ is algebraic over $\mathbb Q$

Prove or disprove the following : $1.\sin ^{-1} 1 $ is algebraic over $\mathbb Q$ $2.\cos (\frac{\pi}{17})$ is algebraic over $\mathbb Q$ As suggested by @Andre ,for the 2nd one ...
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1answer
21 views

Polynomial ring indexed by an arbitrary set.

Let $B$ be any non-empty set, possibly uncountable. What does the term a polynomial ring indexed by the set $B$ means?
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2answers
22 views

Homomorphism of groups

Let $\phi : G_1 \rightarrow G_2$ be a homomoprhism of groups. Now I have to prove that for any $g \in G_1$ we have $\phi (g^{-1})=[\phi (g)]^{-1}$ So how should I begin?
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2answers
31 views

$f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$.

Let $f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$. I'm learning about polynomial rings but my book and my instructor never ...
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2answers
19 views

Group action decomposes $X$ into distinct orbits

Define the group action as $g\cdot x:=g^{-1}xg.$ Let $G=A_5$, and $X=\{\sigma\in A_5:=\sigma=(a,b,c,d,e)\}.$ Show that the group action on X decomposes $X$ into two distinct orbits. There are 60 ...
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1answer
20 views

Representation/Character theory of $S_3$: What is the Vector space $V$?

This is a basic question that I may have a misunderstanding on. When we study the character table of a group, say $S_3$, what vector space are we looking at? I understand that a linear ...
0
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3answers
30 views

Why isn't the number of cosets equal to cardinalities of the groups?

The left coset for subgroup $H$ of $G$ and element $x \in G$ is $$xH=\{xh : h \in H\}$$ Now why is the number of left (or right) cosets not equal to $|G||H|$? Since if one picks $x \in G$ for every ...
6
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1answer
74 views
+50

$R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ Noetherian?

Let $R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ a Noetherian ring ?
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1answer
27 views

Prove that if $J \subset S$ is a principal ideal of $S$, then $f^{-1}(J)$ is a principal ideal of $R$.

Let $R$ and $S$ be commutative rings with unity $1_R$ and $1_S$ respectively, and let $f: R \to S$ be a ring isomorphism . Prove that if $J \subset S$ is a principal ideal of $S$, then $f^{-1}(J)$ is ...
2
votes
1answer
32 views

Confusion with Closures in the topological sense

The rigorous definition is A closure of $A \subseteq X$ of a topological space $X$ is denoted Cl($A$) and is the intersection of all closed subsets of $X$ that contain $A$. The more intuitive or ...
4
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0answers
44 views

Is my proof that $ \mathbb{Q}(\sqrt{5}) $ is the only quadratic subfield of $ \mathbb{Q}(\zeta_5) $ correct?

I would like some feedback on my proof of the fact stated in the question. First, we note that $ L = \mathbb{Q}(\zeta_5) $ is the splitting field of $ X^5 - 1 $, so it is Galois. Furthermore, we know ...
0
votes
1answer
54 views

Don't understand the proof of Artin's “Algebra” Ed 1, Prop 5-8.4

I'm reading Artin's Algebra, Edition 1. In Chapter 5 there's proposition (8.4): Let $c_g$ denote conjugation by $g$, the map $c_g(x) = gxg^{-1}$. The map $f: S_3 \rightarrow Aut(S_3)$ from the ...
0
votes
0answers
23 views

Centralizer of $\sigma\in S_n$ [duplicate]

Let $\sigma\in S_n$ . Describe the centralizer of $\sigma$. Thought: If I conjugate $\sigma$, then $\tau^{-1}\sigma\tau=\sigma.$ This means that $\tau$ is a power of $\sigma,$ and ...