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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Prove $(xa)^{2n + 2} = a^{n + 1}$ if $xax = e$.

$(xa)^{2n + 2} = (xa)^{2n}(xa)^2 = ((xa)^2)^n(xa)^2 = (xaxa)^n(xa)^2 = a^nxaxa = a^{n + 1}.$ I also have two minor problems that don't need their own separate threads. If $xax = b$, then $ab$ ...
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46 views

Showing that the fixed points of a homomorphism form a finite field.

I have the following question on my problem set. Suppose that $\textsf{k}$ is a field and $\phi:\textsf{k}\to\textsf{k}$ is a homomorphism. Check that $\textsf{k}^\phi=\{x\in\textsf{k}:\phi(x)=x\}$ ...
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1answer
55 views

The intersection of subgroup and normal subgroup: the greatest common divisor?

Is the order of intersection of subgroup $H$ and normal subgroup $N$ of group $G$ the greatest common divisor of $\lvert H\rvert$ and $\lvert N\rvert$?
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1answer
136 views

which equations may be solved with compass and a marked ruler?

Ancient Greeks were not able to trisect a general angle with compass and straightedge: now we know that it is impossible, since we would need to solve a cubic equation while only linear and quadratic ...
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3answers
224 views

Find the Galois group of the polynomial

Consider the polynomial $ f(X)= X^4 + 9 \in \Bbb Q [X]$. Show that $f$ is irreducible and find $Gal (K/ \Bbb Q )$, where $K$ is the splitting field of $f$. For the first one I used Eisenstein ...
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0answers
44 views

Show if $R$ is Noetherian, then $R_S$ is Noetherian [duplicate]

Show if $R$ is Noetherian, then $R_S$ is Noetherian. here is what I have read from somewhere else. Suppose $R$ is Noetherian and $J$ is an ideal $R_S$. Then $J=IR_S$ for some ideal $I$ of $R$. Since ...
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1answer
30 views

Does the module $M=\langle x^2+y^2 \rangle$ have a basis over $K[x,y]$

How would I show that the module $M=\langle x^2+y^2 \rangle$ does or doesn't have a basis over $K[x,y]$, where $K$ is a field. Would the set $\{x^2+y^2\}$ be considered a basis for $M$.
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1answer
37 views

Show if $P$ is minimal prime ideal of $R$ then $PR_P$ is the only prime ideal of $R_P$

SHow if $P$ is minimal prime ideal of $R$ then $PR_P$ is the only prime ideal of $R_P$. Here are what I know and don't need to prove: I know $PR_P$ is the maximal ideal of $R_P$. I know $R_P$ is a ...
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0answers
100 views

show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent.

Show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent. The only idea that I come to mind is, we know $PR_P$ is the maximal ideal of $R_P$. Since $P$ is a prime ideal of ...
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1answer
199 views

How many proper nontrivial subgroups do D5 have?

Do I have to find out every element in D5 and draw a table to find out subgroups? I know how to find out every single element in D5, but can't think of how to find proper nontrivial subgroups
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3answers
103 views

Proving subgroups

Question: Let $H$ be a subgroup of $G$ and let $K=\{x \in G: xax^{-1} \in H\ \iff\ a \in H\}$. Prove: (a) K is a subgroup of G So for (a): For closure, I need to show: (i) if $a \in H$ then ...
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2answers
235 views

On groups of four elements

Prove that there is no group consists of 4 elements $a ,b ,c, d$ such that $a^2=c, bc=d$. This is what I understand. For a set $G$ to be a Group with respect to some binary operation $*$, ...
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0answers
156 views

Describe the subgroup of Z generated by 10 and 15.

The book ask the question: Describe the subgroup of Z generated by 10 and 15. So I subtracted 15-10=5 which means the subgroup contains 5. What I'm not understanding is what is the subgroup Z, is ...
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0answers
29 views

Lie Groups map question

This is a question about Exercise 5.59 in Jeff Lee's Manifolds and Differential Geometry. He writes For a fixed $A\in GL(V)$, the map $L_A:GL(V)\rightarrow GL(V)$ given by $A\mapsto A\circ B$ has ...
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1answer
46 views

Show that the right cancellation law holds in $S$

Let $*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x*z = y$. (This $z$ may depend on $x$ and ...
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0answers
93 views

Prove $H \cap K$ is itself a subgroup of $G$ if $H$, $K$ are subgroups of $G$

Let $G$ be a group. If $H$, $K$ are subgroups of $G$, then prove that $H \cap K$ is itself a subgroup of $G$. What I need: I am hoping you wonderful people could read this over and make sure it is ...
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2answers
225 views

Prove that $\{\beta \in S_5 \mid \beta(1)=1, \beta(3)=3\}$ is a subgroup of $S_5$

Let $H = \{\beta \in S_5 \mid \beta(1)=1, \beta(3)=3\}$. Prove that $H$ is a subgroup of $S_5$. How many elements does $H$ contain? Will the conclusion change if you change $S_5$ to $S_n$ for $n ...
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3answers
363 views

Find eight elements commute with one element in S6

Find 8 elements that commute with (12)(34)(56). Do they form a subgroup of S6? I actually have found 8 elements randomly, but I found ab is not in my 8 elements( for some a,b of my 8 elements), so I ...
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1answer
107 views

Subgroup of functions: Show that H is a subgroup of G.

Question: In each of the following, show that H is a subgroup of G. $G=\langle F(\mathbb{R}),+\rangle$, H={ f $\in$ F($\mathbb{R}$): f(x)=0 $\text{ for every }$ x $\in$ [0,1]} I would like someone ...
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2answers
63 views

If $Q$ is a radical ideal of $R$ with $Q∩S=∅$, then $QR_S$ is a radical ideal of $R_S$

If $Q$ is a radical ideal of $R$ with $Q∩S=∅$, show $QR_S$ is a radical ideal of $R_S$. Here is what I have done, since $Q$ is radical, let $q\in Q$ , then $q^n\in I$ for some $I$ ideal. let $a\in ...
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1answer
39 views

Is the localization of R by S is a subset of the ring R

Let $S$ be a multiplicatively closed subset of a commutative ring $R$. Then is it true that the localization $R_S=\{r/s:r\in R, s\in S\}$ a sub-ring of $R$? I think it is true, because $r/s=rs^{-1}$ ...
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2answers
115 views

Biggest splitting field degree given a polynomial of degree n

It's a well know fact that, given $f(x) \in \mathbb{K}[x]$ with $\deg(f) = n$, and being $\mathbb{L}$ its splitting field, we have that $[\mathbb{L}:\mathbb{K}] \leq n!$ What I'd like to know are ...
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3answers
89 views

Compute Galois group and minimal polynomial

Let $ \zeta \in \Bbb C$ be a seventh root of $1$. Find the minimal polynomial of the element $ \alpha = \zeta ^{-1} + \zeta$ over $ \Bbb Q$ and show that if $K= \Bbb Q ( \alpha ) $, then $ K/ \Bbb Q$ ...
4
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1answer
65 views

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$?

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$? (I'm asking this question in order to understand this answer). My idea for a proof so far: $f:=X^{15}-t\in F_5(t)$ is the ...
2
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1answer
35 views

Prove that $U(S) \subseteq U( R)$ (edited with new attempt)!

Let $R$ be a ring with unity which is denoted by $1_R$, and $S$ be a subring of $R$ with unity $1_S = 1_R$ Prove that $U(S) \subseteq U( R)$ Prove that $U(R )$ is closed under multiplication Prove ...
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1answer
40 views

List the elements of $\mathbb Z_2 \times \mathbb Z_3$ and write its operation table(The notaton is additive).

$\mathbb Z_2 \times \mathbb Z_3 = \{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2)\}.$ $$ \begin{array}{c|lcr} + & (0, 0) & (0, 1) & (0, 2) & (1, 0) & (1, 1) & (1, 2)\\ ...
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1answer
44 views

Number of elements in a group ring

Let $R$ be the group ring $\mathbb{Z}_5 S_3$, where $S_3$ is a symmetric group. I need to calculate the number of elements in the group ring, and I'm not sure how to do it, is it just $5^3 = 125$?
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1answer
72 views

Cycle structures of $S_6$

A problem from my algebra homework requests the following: List all the possible cycle structures in $S_6$. For each cycle structure, compute the order of an element with that cycle structure. ...
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0answers
55 views

Prove that $G \times H$ is a group.

$G \times H = \{(x, y): x \in G \text { and } y \in H\}.$ The operation on $G \times H: (x, y) \cdot (x', y') = (xx', yy').$ $(x_1, y_1)[(x_2, y_2)(x_3, y_3)] = (x_1, y_1)(x_2x_3, y_2y_3) = ...
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0answers
20 views

Automorphism of a group [duplicate]

If Automorphhic group of Group G is trivial group then how will you show that order of group G is less than or equal 2.
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1answer
66 views

Commutative ring product of elements in terms of sum of powers of sums

The following fact is stated (but not proved) in Chapter 0 of Noll's Finite Dimensional Spaces (it's 07.21): If $R$ is a commutative ring with unity and $r_1, ..., r_n$ elements of $R$ then $$ ...
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3answers
106 views

Showing that every field is an integral domain.

The proof I have starts of with $\;xy=0\;$ in a field. Then $x^{-1}$ exists because it is a field. Then $x^{-1} xy=x^{-1} 0$. Therefore $y=0$. But surely if an integral domain can not have any ...
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1answer
99 views

If $\varphi(I)$ is an Ideal $\forall I $ ideal of $A$, is $\varphi$ surjective?

Today I heared some young students talking about the fact that if an homomorphism of rings (commutative with identity) $\varphi:A \rightarrow B$ is surjective then the image of any ideal of $A$ is an ...
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1answer
61 views

Principal Ideal Ring and ID

In definition of PID, if we take ring instead of ID call it PIR. I add one more condition: all generators of an ideal are associate to each other. Would it imply PIR with this condition is PID? ...
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2answers
56 views

Is $R$ an integral domain?

Let $R = \{a + b\alpha |\ a,b \in \mathbb{Z}\}\subseteq \mathbb{C}$ where $\alpha = \frac{1}{2}(1+\sqrt{-19})$ Is $R$ an integral domain? To show whether or not $R$ is an integral domain, letting $x ...
0
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1answer
72 views

What is the difference between the representation of a group and an algebra?

Sometimes, I come across this idea in physics -> the representation of Lorentz group: SO(3,1) and the representation of Lorentz algebra: so(3,1). At times, I mix them up. Is there a good intuitive way ...
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1answer
49 views

Is grading unique?

It must be a very basic question, but I just can't figure out... Let $P$ be a graded $A$-module ($A$ is a commutative associative with unity). Can $P$ have two different direct decompositions, that ...
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2answers
47 views

Elementary question on stabilizer and S3

Let $S_3$ be our group. How can I show that $C_G(x) \text{ for }x=(1\,2\,3)$ is $\{1, (1\,2\,3),(1\,3\,2)\}$ without testing every element in $S_3$?
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1answer
263 views

prime implies irreducible

In unique factorization ring with unity(I am not considering commutativity and zero divisor in definition of UFD) irreducible implies prime. And it was proved in ring with unity without zero ...
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1answer
72 views

Module over a polynomial ring

I'm trying to prove the following: If $D = K[x]$ where $K$ is a field, and $M \subset D^n$ is a submodule for some $n \geq 2$ then $L = D^n/M$ is Artinian iff the torsion module of $L$ is isomorphic ...
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4answers
336 views

Identify quotient ring $\mathbb{R}[x]/(x^2-k), k>0$

I need to identify $\mathbb{R}[x]/(x^2-k)$, where $k>0$ (if $k<0$ I believe it's isomorphic to $\mathbb{C}$). If we let $f(x) = x^2-k$, then according to Artin, since $\sqrt{k}$ satisfies ...
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2answers
515 views

Prove that any element in $S_n$ can be written as a finite product of the following permutations

Prove that any element in $S_n$ can be written as a finite product of the following permutations: $(a)\ (12),(13), . . . , (1n)$ $(b)\ (12),(23),...,(n−1,n)$ $(c)\ (12),(12\dots n)$. I have no ...
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0answers
86 views

Show that $a^2 + b^2 = c^3 $ has infinitely many solutions [duplicate]

Show that $a^2 + b^2 = c^3 $ has infinitely many solutions in $ \{ (a,b,c) \in \Bbb Z ^3 | (a,b)=1, (a,c)=1, (b,c)=1 \}$ . Describe all these solutions. I don't know how to approach this question. ...
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2answers
53 views

Let $G$ be an abelian group. Let $\phi$ be a morphism on $G$. Can I say that $\phi(a)\cdot \phi(b)=\phi(b) \cdot \phi(a)$?

My question is this one: Let $G$ be an abelian group. Let $\phi$ be a morphism on $G$. Can I say that $\phi(a)\cdot \phi(b)=\phi(b) \cdot \phi(a)$, with $a$ and $b$ elements of $G$?
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1answer
60 views

Is every finite subgroup of multiplicative subgroup of an integral domain cyclic?

Here is a well-known theorem: Let $F$ be a field and $G$ be a finite subgroup of $F^*$. Then, $G$ is cyclic. Can this theorem be extended to integral domain? Here's a lemma in Weil's elementary ...
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0answers
40 views

Subgroups for ordered pairs

Question: $$G=\langle \mathbb{R}\times \mathbb{R},+ \rangle$$ where $H=\{(x,y)\mid y=2x\}$ is a subgroup of $G$. So is $H$ a subgroup of the group $G$? I started of by checking subgroup conditions: ...
2
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2answers
160 views

Determine whether or not $H$ is a subgroup of $G$ (assume that the operation of $H$ is the same as that of $G$)

I'm trying to answer the question: Determine whether or not $H$ is a subgroup of $G$ (assume that the operation of $H$ is the same as that of $G$). $$G=\langle\mathbb{R}, +\rangle,\ ...
3
votes
2answers
71 views

$\hom (M, \coprod_i N_i) \cong \bigoplus_i \hom (M, N_i)$ in abelian categories when $M$ is simple?

For $M$ a simple $R$ module, and $N_i$ a family of $R$ modules, we have $$ \hom (M, \bigoplus_i N_i) \cong \bigoplus_i \hom (M, N_i) $$ as abelian groups. Since the direct sum is the coproduct in the ...
2
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2answers
317 views

Is $A_n$ non-abelian for $n= 3$?

In the book, it is asked to show that $A_n$ is non-abelian for $n ≥ 4$. Which may imply that it is abelian for $n=3$. Is that so? because $(13)(12)\ne (12)(13)$. Hence is it true to write: $A_n$ is ...
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0answers
103 views

Every odd permutation in $S_n$ can be written as a product of $2n+3$ transpositions?

My question is If Set of permutation set $S_n$ for $n>3$ Prove Every odd permutation in Sn can be written as a product of 2n+3 transpositions and every even permutation as a product of $2n+8$ ...