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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Show that H is dense or isomorphic to Z

Let be S a finite subset of $\mathbb{R}$, and let $H=\langle S \rangle$ the subset generated by S under the sum. a) Let $h_{min}=\inf\{h\in H|h>0\}$. Show that if $h>0$, then h is a element of ...
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1answer
10 views

H is normal subgroup having index $3$, why is every $a^3$ in $H$?

Let $H$ be normal subgroup of a finite group $G$ such that $H$ has index $3$. Show that $a^3$ is in $H$ for every $a$ in $G$. Anyone can help me?
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1answer
11 views

Compute Galois Group of splitting field of $x^p-a$ over $\mathbb Q$

I am having trouble computing the Galois Group of the splitting field $E$ of $x^p-a$ (where $p$ and $a$ are prime) over $\mathbb{Q}$. Let $w$ be a $p^\text{th}$ root of unity, and $\alpha$ a root of ...
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1answer
13 views

“Implicit” condition about separability of a quartic polynomial

Here is an exercise in Hungerford's Algebra, page 277 Ex.12. Let $K$ be a subfield of real numbers and $f \in K[x]$ an irreducible quartic polynomial(of degree 4). If $f$ has exactly two roots, the ...
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1answer
33 views

Are there any system(s) of mathematics whose relationship between variables bears difference to that found within mainstream mathematics?

I have been reading up on boolean algebra quite recently, for those not familiar, this type of mathematical system has much to do with the way logic is represented (and is primarily applied to, though ...
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1answer
23 views

Show that $P = (f(x))$ is a maximal ideal of $F(x)$

Let $F$ be a field and $f(x)$ be an irreducible polynomial in $F[x]$. Prove that $P = (f(x))$ is maximal in $F[x]$. (Here is what I know: $f(x) \neq 0 \wedge f(x) \not \in U(F[x])$, since $f(x)$ is ...
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1answer
28 views

Cardinality of a UFD [on hold]

Why can we be sure that if A is a Unique Factorization Domain and has at least one irreducible element, then A is infinite? I can't see how to prove this. Does it have any connection with primes? ...
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14 views

Showing explicitly that $\alpha \mapsto \alpha+1$ is an automorphism in the Galois group of $x^p-x+a$?

I have already shown that $x^p-x+a$ is irreducible and separable over $\mathbb{F}_p$ and I have shown that if $\alpha$ is a root, so is $\alpha+1$, so $\alpha$ generates the extension, i.e. the ...
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16 views

Prove that $a$ is a prime element of $R$

Let $R$ be a PID and $P = (a)$ is a prime ideal of $R$. Prove that $a$ is a prime element of $R$. Since $P$ is a prime ideal of $R$, let $x,y \in R$ s.t. $xy \in P = (a).$ (WTS $a \mid x$ or $a\mid ...
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2answers
35 views

Show that $S$ is a field

I'm trying to prove the following result: Let $R$ be a principal ideal domain, $S$ an integral domain and $f: R\to S$ a surjective morphism. Prove that if $f$ is not an isomorphism, then $S$ is a ...
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3answers
79 views

A normal subgroup that is not a characteristic

In the book I'm study is written: A normal subgroup of a group need not be characteristic. And as an exercise I'm supposed to find an example, it also said that is pretty hard to find one. After ...
2
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2answers
56 views

What is an irreducible polynomial in $\mathbb{Z}$ that has root $\sqrt{2}+\sqrt{3}$?

What is an irreducible polynomial in $\mathbb{Z}$ that has root $\sqrt{2}+\sqrt{3}$? Obviously that root is not in $\mathbb{Z}$. I tried ...
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18 views

Matrix representations and idempotent/nilpotent elements

A conceptual question: Let's say there's a linear matrix representation of a particular algebra. I'm wondering just how much this matrix representation can tell us about the 'outstanding' elements ...
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1answer
23 views

Ring Homomorphism Textbook Question

Please help me understand the last three sentences in this paragraph from the Artin textbook. Where does this come from: "The monomials that appear in $r_0(t^2)$ have even degree, while those in ...
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0answers
19 views

$T(M) \cong R<x_1,\ldots,x_n>$ isomorphism question

I have that $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$ where $T^k(M) = \bigotimes_{i =1}^{k} M$. In a paper i have that to prove such isomorphism, we define: $$\Phi:R<x_1, \ldots, x_n> ...
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0answers
24 views

Every module over a division ring is free?

I am currently trying to answer the following true/false question: True or False: Every module over a division ring $R$ is free. I know the result would be true if $R$ is a field (ie a ...
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31 views

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ [on hold]

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ is called capable. I want to know which one of the following groups is capable? $(\mathbb{Z}_2\times D_8)\rtimes \mathbb{Z}_2$, ...
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4answers
47 views

Are primary ideals always contained in unique maximal ideal?

Just wondering, is this a standard fact? I notice a couple Banach algebra texts define primary ideals in this way. Another question: does this property, i.e. being contained in a unique maximal ideal, ...
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0answers
22 views

Tensor algebra becomes a graded $R$-algebra short proof verification

I had a post proving that the tensor algebra becomes a graded ring, i have come up with a simple approach that goes as follows: Proposition: The tensor algebra $T(M)$ with multiplication defined ...
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24 views

Is the discriminant of a polynomial surjective onto $\mathbb Z$?

Consider polynomials of degree two over $\mathbb Z$: $f = ax^2+bx+c$ The discriminant is $D = b^2-4ac$ And we can show that $D=2$ is not a possible value for $D$. I wonder if the value $D=2$ ...
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2answers
34 views

A given ring of matrices has an infinite number of invertible elements

The set $\mathcal{M} = \bigg\{ \begin{pmatrix} a & 2b \\ b & a \\ \end{pmatrix} \bigg\vert a,b \in \mathbb{Z} \bigg\}$ is given. Prove that: (1) $\mathcal{M}$ is a commutative ring with ...
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2answers
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how to find subgroup generated by elements [on hold]

Let $G=\mathbb{Z}_6$ the cyclic group of order $6$ then $\langle 2\rangle=\{0,2,4\}$ but $\langle 2,3\rangle=\mathbb{Z}_6$. Can someone help me with this? I have a confusion on $\langle ...
2
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1answer
36 views

Least common multiple for integer matrices

Given two full-rank $3\times3$ integer matrices $M_1$ and $M_2$, I am trying to find integer matrices $N_1$ and $N_2$ such that $M_1N_1$=$M_2N_2$, such that $\left|\det(M_1N_1)\right|$ is minimal. ...
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1answer
30 views

Can this property of the Gaussian integers apply to other rings like $\mathbb{Z}[\sqrt{2}]$?

The ring $\mathbb{Z}[i]$ has the property that an element of the ring is irreducible if its norm is a prime number congruent to 1 (mod 4). Also, it is irreducible of the elements are associates of the ...
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3answers
58 views

Finite dimensional $\mathbb R$-algebras without zero divisors

Let $A$ be a finite dimensional commutative algebra over $\mathbb{R}$ without zero divisors. Prove that $\dim A = 2$. Because any $a \in A\setminus \{0\}$ is not a zero divisors, thus $ab \neq 0$ ...
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1answer
42 views

Is the direct product $\mathbb{Z} \times \mathbb{Z}$ a cyclic group, with the operation $(x, z) + (y, a) := (x + y, z + a)$? [on hold]

Is the direct product $\mathbb{Z} \times \mathbb{Z}$ a cyclic group, with the operation $(x, z) + (y, a) := (x + y, z + a)$? I really do not know how to this question, any hints or a hopeful step by ...
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2answers
45 views

Is there an isomorphism of additive groups when $\mathbb{Q/Z}$ isomorphic to $\mathbb{Q}$? [duplicate]

I know that I have to study the order of every element in $\mathbb{Q/Z}$. But what do I do? I've been struggling of what to do for this question
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2answers
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In which algebraic setting can I state (and prove) the binomial theorem?

In a book on algebra I'm currently working with a proof that uses the binomial theorem for $(x+y)^m$ where $x,y$ are elements of some arbitrary field $k$. This looks strange to me, so I did some ...
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1answer
57 views

How to define the isomorphism?

Let $R$ be a ring, then For $R[x]/\langle x-1\rangle \cong R$, we define the map, $\varphi$ : $R[x]\rightarrow R$, defined by $\varphi(f) =f(1)$ For $R[x]/\langle x\rangle \cong R$, we define the ...
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1answer
38 views

Characteristic of a field [on hold]

If $K$ is an infinite field, then $Char K = 0$ but the reverse is not sure. Examples of $Char K = 0$ but not infinite field $K$?
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1answer
47 views

There exists a homomorphism $f : G \to H$ with $|G| = 20$ and $|im f | = 6$

There exists a homomorphism $f : G \to H$ with $|G| = 20$ and $|im f | = 6$? Is this true? I know that I have to use the first isomorphism theorem but I don't know what to do next?
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0answers
17 views

Quotient of graded rings by graded ideals are again graded rings

I want to prove the following statement. Let $A = \bigoplus_{n=0}^{\infty}A_n$ to be a graded $R$-algebra and $I$ a graded ideal of $A$. Let $(A/I)_n = (A_n + I)/I$ be the image of $A_n$ in $A/I$. ...
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1answer
24 views

Boolean algebras and rings

I know that M. H. Stone proved that there is a bijection between boolean algebras and boolean rings. The bijection I know is the following: to any given Boolen algebra $(L,\, \vee, \wedge)$ we ...
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19 views

R has IBN but R fails rank condition [on hold]

I need an example about "IBN for ring": R is a ring (no commutative), R has IBN but R fails rank condition. thanks
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0answers
37 views

Tensor algebra becomes an R-algebra. Theorem, Proof, Dummit and Foote

I have the definition of tensor algebra as follows: $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$, where $M$ is an $R$ module, where $R$ is commutative and contains the element $1$. Finally $T^k(M) = ...
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0answers
16 views

In any integral domain, only $1$ and $-1$ are their own multiplicative inverses.

In any integral domain, only $1$ and $-1$ are their own multiplicative inverses. Note that $x=x^{-1}$ iff $x^2=1$ I'm not sure how to go about proving this. I know the definition of an ...
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1answer
26 views

Image and kernel of the natural projection from a group to its quotient by a normal subgroup [on hold]

Let $H$ be a normal subgroup of a group $G$. Prove that the function $$f : G \to G/H$$ defined by $f (g)=gH$ is a homomorphism with image $G/H$ and kernel $H$. Use this fact to conclude that a ...
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2answers
39 views

Why is F($\beta$) a subfield of F($\alpha$)?

There is a corollary in my book that says: If E is an extension field of F, $\alpha \in E$ is algebraic over $F$, and $\beta \in F(\alpha)$, then $\deg(\beta,F)$ divides $\deg(\alpha,F)$. In ...
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1answer
44 views

How can one find irreducible elements in $\mathbb{Z}[\sqrt{2}]$?

Is it just all elements that have a prime norm? Ok, so from the comment I've seen that elements with a non prime norm can be irreducible...what exactly do I have to do to find irreducibility?
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1answer
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Is this generalization of Boolean algebras a variety?

I'm interested in the class of structures $\langle S,\top,\neg,\wedge\rangle$ defined by the axioms: $p \wedge q=q \wedge p$ $p \wedge (q \wedge r) = (p\wedge q)\wedge r$ $p = p \wedge p$ $p = ...
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1answer
15 views

Ring polynomial kernel generators

This is the textbook question: Q: Find generators for the kernels of the following maps: $\mathbb{R}[x,y] \to \mathbb{R}$ defined by $f(x,y) \mapsto f(0,0)$ $\mathbb{R}[x] \to \mathbb{C}$ defined ...
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1answer
51 views

Can one define alternative algebras? [on hold]

Context I am an engineer by profession, and have read up on math some - where I got the sense that it was common to define alternative algebras for specific domains - where you define the operators, ...
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50 views

Sylow p-subgroups and set X not divisible by p

Let $P$ be a Sylow $p$-subgroup of $G$ and suppose that $P\subseteq Z(G)$. Show that the set $X$ of elements of $G$ with order not divisible by $p$ is a subgroup of $G$ and that $G=P\times X$. I ...
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1answer
45 views

Isomorphism of Localizations

I believe, though a not sure, that any two ideals $A, B$ of a Dedekind domain $X$ are isomorphic as $X$-modules iff their localizations $A_p, B_p$ are isomorphic for any prime ideal $p$. Could anyone ...
2
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2answers
36 views

prove the nomalizer $N(H)$ of the subgroup $H$ in $G$ is a group

I need some help on the following question. For an arbitrary subgroup $H$ of the group $G$, the normalizer of $H$ in $G$ is the set $N(H) = \{x \in G \mid xHx^{-1} = H\}.$ Any help??
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1answer
28 views

If $P$ is a prime then $R/P$ is an integral domain.

I know the same question has been already asked here. So, I am not asking for any proof rather to find out what's wrong with my proof. So, this is what I did: Let, $a+p, b+p \in R/P$, since $P$ is a ...
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0answers
22 views

Quotient Groups with symmetric $S_4$ group [duplicate]

I'm working on this problem and I am having trouble figuring it out. The problem is: Find the quotient group $G/H$. Write out the distinct elements of $G/H$ and construct a multiplication table of ...
3
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0answers
27 views

Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$

Let $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$ Suppose $f(x)$ and $g(x)$ are relatively prime in ...
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1answer
33 views

Show that it is a homomorphism?

For any abelian group $G$ we have $e_n: G \to G, e_n(g) = g^n$. By convention $e_0(g) = 1$. For a Field $F$ we have the subgroup $\{1,-1\} \leq F^*$. When $F$ is of characteristic $2$, this is the ...
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0answers
12 views

Prove that on N, the relation V where mV n

This is my question! Help me, please! Prove that on $\Bbb N$, the relation $\mathsf V$ is a linear order where $m\mathsf Vn$ if and only if $m$ is odd and $n$ is even, or $m$ and $n$ are even and ...