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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5
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1answer
38 views

$1+\frac{1}{2} +\frac{1}{3} +…+\frac{1}{p-1} =\frac{a}{b}$

Let $p\gt 3$, be a prime number and $1+\frac{1}{2} +\frac{1}{3} +...+\frac{1}{p-1} =\frac{a}{b}$ when $a,b\in \mathbb N$ and $gcd(a,b)=1$. prove that $p^2|a$. I proved that $p|a$, but I cant ...
0
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0answers
8 views

What is the degree of n over a field F if we let v be algebraic?

I have a final coming up in a few hours and I can't seem to solve this practice problem. If anyone could lay down some ground work to get me started that would be great or solving it all the way would ...
-1
votes
0answers
32 views

Prove they are not pairwise isomorphic [on hold]

Prove $\mathbb{Z}_8$, $G_s$(the group of symmetries of the square) and the quaternion group $Q$ are not pairwise isomorphic. Stuck on this question. Seems very difficult.
1
vote
2answers
35 views

Commuting of Hom and Tensor Product functors?

Let $V_i,W_i$ be finite dimensional vector spaces, for $i=1,2$. Assume we have homomorphisms $\phi_i:V_i\rightarrow W_i$. Then, there is an induced map $\widehat{\phi_1 \times \phi_2} \in Hom(V_1 ...
0
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0answers
41 views

Character group of $\mathbb{Z}$

I am trying to compute the character group of $\mathbb{Z}$ which contains homomorphisms that map into $\mathbb{C}^\times$. I have determined that each homomorphism $\phi \in \hat{\mathbb{Z}}$ may be ...
0
votes
0answers
10 views

Prove that in cyclic codes, ($C_1$+$C_2$)$^\perp$=$C_1^\perp$+$C_2^\perp$

Let $C_1$ and $C_2$ be cyclic codes over finite field with the same length. Prove that ($C_1$+$C_2$)$^\perp$=$C_1^\perp$+$C_2^\perp$. The direct conclusion is clear but how to prove the reverse ...
0
votes
2answers
37 views

$\mathbb K[s, t]/\langle s-t\rangle\simeq \mathbb K[t]$ as $\mathbb K$-algebras?

How can I show there is an isomorphism of $\mathbb K$-algebras (where $\mathbb K$ is a field): $$\mathbb K[s, t]/\langle s-t\rangle\simeq \mathbb K[t]?$$ Above $\mathbb K[s, t]$ and $\mathbb K[t]$ ...
2
votes
3answers
41 views

Irreducibility of polynomials over finite field of integers $\bmod 11$.

Theorem (Fermat's little Theorem). If $p$ is a prime and $a \in \mathbb{Z}$ with $a \nmid p$ then $a^{p-1} \equiv 1 \mod p.$ Let $\mathbb{Z}/p\mathbb{Z}$ denote the multiplicative group of ...
6
votes
6answers
112 views

In a ring with no zero-divisors, for $(m,n) =1$, $a^m = b^m$ and $a^n = b^n$ $\iff a =b$

Let $R$ be a ring with with no zero divisors. If $a, b \in R$ are such that $a^m = b^m$ and $a^n = b^n$, where $m$ and $n$ are relatively prime positive integers, then show that $a = b$. My ...
2
votes
1answer
13 views

Discriminant of a product of polynomials

Let $f,g$ be irreducible, monic and in $\mathbb Z[x]$. Then (I hope this is correct) $disc(f\cdot g)=disc(f)\cdot disc(g)\cdot\prod_i\prod_j(a_i-b_j)^2$ where the $a_i$ are the roots of $f$ ...
0
votes
2answers
17 views

Example of formally real field $F$ with irreducible polynomial of odd degree in $F[x]$

Let $F$ be a formally real field; i.e. $-1$ cannot be expressed as the sum of squares. Let $K/F$ be a field extension of odd order. Hence, there exists $\alpha\in K$ which satisfies an irreducible ...
0
votes
1answer
26 views

Prove that $|V| = |F|^{\dim V}$ for a finite dimensional vector space $V$ over $F.$

In my algebra class, we proved that a quotient ring $F[x]/(f(x))$ is a vector space over $F$ and $\dim_F F[x]/(f(x)) = \deg f(x).$ I am attempting to use these facts to prove that the field $U = ...
5
votes
2answers
54 views

For what Abelian Group $A$ is there the following exact sequence?

The question is: For what kind of Abelian group $A$ is there a short exact sequence: $$ 0\to\mathbb{Z}\to A\to\mathbb{Z}_{n}\to0. $$ This is an exercise from Allen Hatcher's Algebraic Topology ...
0
votes
1answer
19 views

he first isomorphism theorem to deduce that G/K ⇠= H.

For an abelian group $G$, consider the sets $H = \{g^2 | g\in G\}$ and $K = \{g\in G | g^2 = e\}$. $f : G \to G$ defined by $f(g) = g^2$ is a homomorphism. use the first isomorphism theorem to ...
-4
votes
1answer
48 views

Sum of elements of a finite field is zero [on hold]

How to prove the sum of the elements of the finite field is zero ?
4
votes
1answer
29 views

Galois Extension whose Galois Group is $\mathbb{Z}_2\oplus\mathbb{Z}_4$

The book I am using for my Abstract Algebra course is Contemporary Abstract Algebra by Joseph A. Gallian. Let $E/F$ be a Galois extension with Galois group isomorphic to ...
3
votes
4answers
313 views

Definition of prime element in a Euclidean ring does not make sense. Herstein - Topics in Algebra

Herstein's Definition: In the Euclidean ring $R$, a nonunit $\pi$ is said to be a prime element of $R$ if whenever $\pi=ab$, where $a,b \in R$, then one of $a$ or $b$ is a unit in R. $\mathbb Q$ is a ...
0
votes
0answers
20 views

Show that H is dense or isomorphic to Z [on hold]

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 ...
0
votes
1answer
27 views

H is normal subgroup having index $3$, why is every $a^3$ in $H$? [on hold]

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?
0
votes
1answer
18 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 ...
0
votes
2answers
24 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 ...
0
votes
1answer
35 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 ...
2
votes
1answer
26 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 ...
1
vote
1answer
31 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? ...
1
vote
1answer
28 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 ...
0
votes
0answers
20 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 ...
1
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2answers
36 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 ...
2
votes
3answers
88 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 ...
4
votes
2answers
69 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 ...
0
votes
0answers
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 ...
0
votes
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 ...
2
votes
0answers
26 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> ...
3
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0answers
29 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 ...
1
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0answers
37 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? $(\mathbb{Z}_2\times D_8)\rtimes \mathbb{Z}_2$, ...
0
votes
4answers
55 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, ...
0
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0answers
26 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 ...
2
votes
0answers
35 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 ...
0
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2answers
35 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 ...
-2
votes
2answers
34 views

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
votes
1answer
37 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. ...
2
votes
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 ...
1
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3answers
61 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 $ab \neq 0$ ...
-1
votes
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
46 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
14
votes
2answers
641 views

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 ...
1
vote
1answer
58 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 ...
-3
votes
1answer
39 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 $K$?
1
vote
1answer
50 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?
0
votes
0answers
18 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 $A/I$. ...
0
votes
1answer
25 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 ...