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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Multiplicative order of $5$ in $(\mathbb{Z}/101\mathbb{Z})^{*}$?

I have to find the multiplicative order of $5$ in the group $(\mathbb{Z}/101\mathbb{Z})^{*}$. Any help? Thanks!
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10 views

How to calculate the cardinality of $\mathbb{Z}[\sqrt{-17}]/(3, 1+\sqrt{-17})$ and $\mathbb{Z}[\sqrt{-17}]/(\sqrt{-17})$?

thanks for taking the time to look at my problems. I was trying to calculate the norm of $(3, 1 + \sqrt{-17})$ and $(\sqrt{-17})$. The second one is 17 because of the norm of the element ...
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14 views

Zeros of specialization of a family of polynomials

Let $k$ be an algebraically closed field, and $K\supset k$ be an algebraically closed extension. Let $a\in K^n$ be a tuple, we call $a^\prime\in k^n$ a specialization of $a$ if for any $f(X)\in k[X]$ ...
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14 views

Constraint for matrix representation for general irreducible permutation group.

Say I have a matrix $\bf P$ for which is ensured that $P_{ij} = \{0,1\}$. Then consider this requirement: $$\sum_{k=0}^{n-1}{\bf P}^k[1,0,\cdots,0]^T = [1,1,\cdots,1]$$ Should this be enough to make ...
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2answers
16 views

Is a sub-algebra of a semisimple algebra nescesarily isomorphic to a subset of its direct product decomposition via Wedderburn's classification?

Wedderburn's classification of semisimple algebras tells us that any semisimple algebra $A$ is isomorphic to a finite direct product of matrix algebras over division algebras, say $A \cong ...
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Is $\operatorname{Spec} O_K$ regular?

Consider a number field $K$ and let $O_K$ be its ring of integers. Is $\operatorname{Spec} O_K$ a regular scheme? In other words I'm asking if for every prime ideal $\mathfrak p$, then ...
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15 views

Normal extension of a field

Let $F$ be an extension of $K$ (they are both fields). I know that if $F$ has finite degree over $K$, then the following things are equivalent: 1) $F$ is a normal extension of $K$ 2) If $T$ is an ...
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1answer
17 views

On subring of a Field

Let $R$ ba a subring of a field $F$ such that for each $x \in F$ either $x \in R$ or $x^{-1} \in R$. Prove that If I and J are are ideals of R, then either $I \subseteq J$ or $J \subseteq I$.
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13 views

Show that an $m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$

Show that an $m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$. I was thinking of mapping rows to columns of the matrix. Let $$M = \begin{bmatrix} ...
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1answer
11 views

Question concerning modules over a clifford algebra

Let $R$be a commutative ring with unit element, and $M$ be an $R-$module. Let $f:M \times M \to R$ be a nondegenerate symmetric bilinear quadratic form, and $C(f)$ be the corresponding clifford ...
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28 views

Is there a way to characterize the prime ideals in $\mathbb{R}[x_1,x_2, \dots , x_n]$?

I'm studying algebras which can be formed by the quotient of principal ideals in $\mathbb{R}[x_1, \dots , x_n]$, and thus would like to be able to determine which of said principle ideals are maximal, ...
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1answer
67 views

Is 2 is prime in $\mathbb{ Z}_6$?

Prove that $2$ is prime element in $\mathbb{ Z}_6$? I have proved it using Caleys Table, but can someone suggest a theoretical method ?
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1answer
24 views

Polynormal subgroup

Let $G$ be a group. $H$ is said to be polynormal in $G$ if for each $x\in G$, we have $H^{\langle x \rangle} = H^{H^{\langle x \rangle}}$ where $H^{\langle x \rangle} = \langle x^nHx^{-n} \;|\; n\in ...
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Show that the product of two transpositions can be expressed as a product of $3$-cycles

Consider the symmetric group $S_n$ where $n>2.$ Show that the product of two transpositions $(ab),\,(cd)$ can be written as a product of $3$-cycles where $a,b,c,d$ are all distinct. I'm not sure ...
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7 views

A modularity condition

Let $R\subseteq S$ be rings with unity and $X$ ,$Y$ be subsets of $S$ with $X$ an ideal. If $S=X+Y$ what conditions should be held to infer the equality $R=(X∩R)+(Y∩R)$? I think that if we ...
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8 views

Isomorphism classes and invariant factors of abelian group

Let $G$ be an abelian group with $ord(G)=3374=2\cdot 7\cdot 241$. Calculate all isomorphism classes with the invariant factors $k_1\ ...k_n$ sucht that $k_i$ divides $k_j$ $(i<j)$. Since ...
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2answers
43 views

Prove that $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(i \sqrt{5})$ are not isomorphic.

The question is : Prove that $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(i \sqrt{5})$ are not isomorphic (I'm talking about ring isomorphism). What I have done : suppose there is an isomorphism ...
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0answers
19 views

Is every mondule homomorphism $\mathbb{Z}\to\mathbb{Z}$ of the following form?

Let $\psi:\mathbb{Z}\to\mathbb{Z}$ be a module homomorphism. Is it true, that $\psi(n)=kn$ for some $k\in\mathbb{Z}$?
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Prove that $M_p$ is a ideal of $\mathbb Z/(p)[x]$ and $\mathbb Z[x]/M$ is isomorphic to $\mathbb Z/(p)[x]/Mp$.

Let $M_p$ = $\gamma (M)$, the image of $M$ ($M$ is a maximal ideal of $\mathbb Z [x]$) in $(\mathbb Z/(p))[x]$, where $\gamma$: Z[x] --> Zp[x] is the morphism such that $\gamma (\sum_i a_ix^i)=\sum_i ...
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17 views

The standard tropical lines - Pointwise valuation

Let $K$ be an algebraically closed field with valuation, and its value group $\Gamma_\text{val}^n$ is dense in $\mathbb{R}$. Consider the polynomial $ f(x,y)=x+y+1. $ It can easy by shown that the ...
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1answer
17 views

$[F(a,b):F(a)]=[F(b):F]\iff F(a)\cap F(b)=F$

Let $F$ be a field and $a,b$ be elements of some algebraic extension of $F$. Is it true that $[F(a,b):F(a)]=[F(b):F]\iff F(a)\cap F(b)=F$? I have a proof for the forward implication: Let $c\in ...
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1answer
16 views

Showing an Ideal is the ring

If A is an Ideal of a ring R and the unity 1 belongs to A, prove that A=R. It is a sufficient condition to show that $A\subseteq R$ and $R\subseteq A$. Indeed, it is trivial to see that ...
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1answer
44 views

If $e$ and $f$ are idempotent and $e(e+f)f=e+f$, then prove that $e=f$

Let $R$ be an associative ring and let the idempotents $e$ and $f$ belongs to $R$. Then prove that $$e(e+f)f=e+f \iff e=f$$ The only if part is very easy and I have proved what about the ...
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18 views

Singular ideals and rings

In Lam's book, Corollary (7.4)(2) says that for a nonzero ring $R$ we have $Z(R_R)≠ R$, where $Z(R_R) $ stands for the singular ideal of $R$.. But, some nonzero commutative rings are "singular" in the ...
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35 views

A finite group theory problem [duplicate]

Let G be a finite group such that $a^{2}b^{2} = b^{2}a^{2}$ and $a^{3}b^{3} = b^{3}a^{3}$ for all $a,b\in G$. Prove that $G$ is abelian. I was wondering if there is any other elegant and general ...
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Hall $\pi$-subgroups of normal subgroups

Let $A$ be a normal subgroup of $G$ such that $H\in$ Hall$_\pi$($A$) and $G/A$ is a $\pi$-group. Suppose that $H = H_1 \cap A$ where $H_1 \in$ Hall$_\pi$($G$). Show that $H^A = H^G$ where $H^G = ...
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1answer
25 views

Intersection of any set of ideals is an ideal

Prove that the intersection of any set of Ideals of a ring is an Ideal. I'm looking for hints. Let A, B both be Ideals of a ring R. Suppose $I \equiv A\cap B$. Since A and B are both Ideals of ...
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25 views

Proof Verification of Result Involving Maximal Ideals

In further investigation of a question I asked earlier, I came across the following result, the proof of which I hope can be looked over here. I personally find it kind of interesting and I hope ...
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12 views

Text introducing $T^{i,j}$-tensor algebra

I'm reading a lecture note here : http://www.cis.upenn.edu/~cis610/diffgeom7.pdf It introduces $T^{•,•}(M)$ the tensor algebra and says that this is a necessary tool in differential geometry. Well, ...
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1answer
14 views

Proving that something equals the commutator subgroup and conjugacy classes/normal subgroups

I've learned that the commutator subgroup is generated by the commutators. Now this says little about its elements (to me) because I don't see how they need to be commutators themselves. I'm ...
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19 views

Find $T_1(\langle (1,2,3,4,5,6,7,8,9) \rangle )$

$T_k(G)=\{g \in G : o(g) \, \,|\, \, p^k \}$ I am unsure what to do. Let that long permutation be $b$. Do we just find calculations of $b$ like $b^2$ or $b^{-1}$ or $b^b$ etc, that would give an ...
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1answer
29 views

Show $T_k(G)=G$

Suppose $G = C_n$ is an abelian $p$-group (so that $n = p^a$ for some $a$). Show that $T_k(G) = G$ if $a ≤ k$ and $T_k(G) = C_{p^k}$, otherwise. $T_k(G)=\{g \in G : o(g) \, \,|\, \, p^k \}$ If $a ...
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1answer
14 views

Prove $T_k(G)$ is a subgroup

$G$ is an abelian $p$-group. $T_k(G)=\{g \in G : o(g) \, \,|\, \, p^k \}$ for $k \ge 0$. Prove that this is a subgroup. For closed within multiplication, it is easy to see that $o(a)o(b)\,\, | \, ...
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1answer
39 views

Show that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$ [duplicate]

Let $f: A \to B$ and $g: B \to C$ be invertible mappings; that is, mappings such that $f^{-1}$ and $g^{-1}$ exist. Show that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$. I tried using the facts that ...
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2answers
27 views

Finding the order of an element

When we have permutation elements like $b=(12)(234)(1223)$ we can easily say that the order of each cycle is $2$, $3$ and $4$ respectively so the order of $b=\text{lcm}(2,3,4)$. When we have $C_n$, ...
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53 views

Subgroups of $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$

If I'm understanding the main theorem of (infinite) Galois theory correctly, applied to $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$, it gives us: a) all its open subgroups are $Gal(\bar{\mathbb{Q}}/K)$, with ...
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1answer
47 views

How can we find the elementary divisors?

We consider the polynomial ring $R=\mathbb{R}[t]$ and the $R$-module $M=\mathbb{R}^3$, where $a\cdot x$ ($a\in R,x\in M$) is defined as usual if $a\in \mathbb{R}$, and $a\cdot x=(x_1, 0, 0)$ if ...
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Finding subgroups of $D_8$

$$D_8=\{(),(1234),(13)(24),(1432),(13),(24),(14)(23),(12)(34) \}$$ Am I right to say that to find the subgroups, we have to make sure that the identity can be generated or is in the group and the ...
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1answer
30 views

Isomorphism of group algebras $k[G\times H]\cong k[G]\otimes k[H]$ and interpretation

could someone check it the following is correct? I want to show the isomorphism of $k$-modules ($k$ a Ring) as mentioned in the title. I would like to simplyfy the situation to two finite groups ...
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2answers
28 views

Isomorphic normal subgroups [duplicate]

Let $N_1$ and $N_2$ be normal subgroups of G so that $N_1$ and $N_2$ are isomorphic. Is it true that then also $G/N_1$ is isomorphic to $G/N_2$?
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32 views

Characters defined by cyclic extensions

Let $F$ be a finite cyclic extension of degree $p$ over ${\bf Q}$. As I understand it, there is a way to associate a cyclic character to this extension. How does one do this explicitly? And how far ...
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$C_2 \times C_2$ and $C_4$

With the cyclic group $C_n $, why can't we say $C_4$ is isomorphic to $C_2 \times C_2$? Is it because 2 and 2 are not coprime? Are there any other types i should watch out for?
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1answer
45 views

Is there an Unique Generating Set of a Group when Cardinality of the set is fixed?

$A$ is a generating set of permutation-group $G$ (not a p-group), i.e. $\langle A \rangle=G$. $|A| \leq \log_2(|G|)$. $A$ has possible maximum number of elements to generate $G$. It means that the ...
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Proof that the quotient of equal rank free abelian groups is finite?

Let $G$ be a free abelian group with finite rank and $H$ be a subgroup of $H$ such that $rank(G)=rank(H)$. Then, why is $G/H$ finite? Since $G/H$ is finitely generated, there is a free abelian group ...
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Characteristic of a Ring definition [on hold]

I read the definiton of characteristic of a ring and it doesn't make sense to me 1+1 ...+1(n times) =0 ,we sum the integer 1 n times and we want it to be 0 or there is a different idea I can't get it ...
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Show that if $\operatorname{char}(R) = m$ and $\operatorname{char}(K)=n$, then $\operatorname{char}(R \times K) = \operatorname{lcm} (m,n)$

Show that if $\operatorname{char}(R) = m$ and $\operatorname{char}(K)=n$, then $\operatorname{char}(R \times K) = \operatorname{lcm} (m,n)$. I don't know where to start....
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108 views

What field of maths is this?

Suppose I have a simple function $f(n,x)$ which for some integer $x$ generates a unique rational number $f$ for each $n\in\mathbb{N}$ - for argument's sake lets imagine it's $\frac{x^2}{5n}$. Suppose ...
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1answer
21 views

Determining the degree of a root of unity over a cyclotomic expansion

For $\xi_{n} = e^{2\pi*i/n}$ , determine the degree of $\xi_{7}$ over the field $\Bbb{Q}(\xi_{3})$ How would I approach this problem? I'm having trouble starting this problem and can't find any ...
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23 views

Prove: $\bar{a}^2 = \bar{0}$ in $\mathbb{Z}_{pq} \rightarrow \bar{a}=0$ where $p\neq q$ are primes

For this summer, I am teaching myself abstract algebra and I've been working on a proof for the following statement. I just need someone to confirm whether it is sound. (Note: Here, $\bar{a}$ denotes ...
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3answers
27 views

Permutation and order

I know that the order of a group is the number of the elements, then if we have a permutation what does the order of permutation mean? The number of distinct elements?