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.

learn more… | top users | synonyms (1)

2
votes
1answer
49 views

Is $\mathbb{Z}[\sqrt{-7}]$ a UFD or not? [duplicate]

Is $\mathbb{Z}[\sqrt{-7}]$ a UFD or not? I feel like there should be some easy example showing that it is not, but I can not think of one.
0
votes
0answers
35 views

Quotient Ideal, quick equivalence.

Define $(I:J) = (r \in R : rJ \subset I)$. Show $IJ \subset K \iff I \subset (K:J)$. Show the ideals $J,I,K$ are ideals of $\mathbb{Z}$, then $(< x >:<y>) = <z>$. where $x = yz$. I ...
0
votes
1answer
18 views

Polynomial ring ideal. Show that $(f) \cap R_N = (0) \iff f \notin R_N$

I have a polynomial ring $R = K[x_1,\dots,x_n]$ and $N \subset \{x_1,\dots,x_n\}$. Define $R_N = K[x_i:x_i\notin N]$. If $f\not= 0$, show $(f) \cap R_N = (0) \iff f \notin R_N$. So my entire argument ...
0
votes
1answer
81 views

Algebraic element - integral domain

Let $K$ a field and $L$ a subfield of $K$. Let the set $\overline{L}:= \{k \in K: k$ is algebraic over $L$ $\}$ is another subfield of $K$. Show that $\overline{\overline{L}}=\overline{L}$. ...
0
votes
0answers
36 views

$R[x]$ has a subring isomorphic to $R$ [duplicate]

$R$ is a commutative ring. We need to prove that $R[x]$ has a subring isomorphic to $R$. Let $S$ be that subring of $R[x]$ which has polynomials of even degree. Now I consider a mapping from $S$ ...
0
votes
1answer
25 views

Prove that the Gaussian integer $a$ is a prime element if $N(a)=p$ or $p^2$ where $p$ is congruent t0 3 mod 4

Let $a \in \mathbb{Z}[i]$ such that $N(a)$ is a prime or the square of a prime congruent to 3 modulo 4 in $\mathbb{Z}$. That is, $N(a)=p$ or $p^2$ where $p \equiv 3 \bmod 4$. Prove that $a$ is a ...
0
votes
1answer
70 views

Do there exist polynomials $f,g$ in $\mathbb{C}[x]$ such that $(x^2 - 1)f + x = g^2$.

Do there exist two polynomials $f, g \in \mathbb{C}[x]$ such that $(x^2 - 1)f + x = g^2$? I know that this cannot happen in $\mathbb{R}[x]$. However, since $\mathbb{C}$ is algebraically closed, this ...
0
votes
1answer
26 views

A two-variable quadratic form over a field of characteristic 2 with no nontrivial roots

I'm looking for a quadratic form of the form $q(x,y)=ax^2 + bxy + cy^2 \in F[x,y]$, where $F$ has characteristic 2, and $q(x,y)$ has no roots besides the obvious one, $x=y=0$. I've proved the case of ...
1
vote
3answers
27 views

A nontrivial solution to the quadratic form $x^2 - xy + y^2$ over the finite field $𝔽_p$ with $p ≡ 1 \pmod3$ a prime

I'm trying to prove that when $p ≡ 1 \pmod3$ is a prime, $p$ is reducible over the Eisenstein integers, and I've gotten to the point where, provided $p\,|\,u^2 - u + 1$ for some integer $u$, then $p$ ...
0
votes
2answers
48 views

Two ways to decompose $\mathbb Q[G]$ for $G = \{1,g,g^2\}$ and their interrelation.

Let $G = \{1,g,g^2\}$ be a cyclic group of order $3$. Consider the group ring $\mathbb Q[G]$. Then we have the two $G$-invariant simple subspaces $$ I = \{ a_0 + a_1 g + a_2 g^2 : a_0 = a_1 = a_2 \} ...
1
vote
1answer
36 views

Show that if $H$ is a normal subgroup of $G$ then so is $\bar{H}$.

This is problem 4.14 in Armstrong's Basic topology: Let $G$ be a topological group. If $H$ is a subgroup of $G$, show that its closure $\bar{H}$ is also a subgroup, and that if $H$ is normal then ...
0
votes
1answer
35 views

Quotient of maximal and prime ideals [on hold]

Given that $I, J$ are ideals in $R$, $I$ is maximal or prime, do we have that $I/J$ is maximal in $R/J$? $I/J$ is prime in $R/J$? I think it is true but don't see how it works.
2
votes
3answers
70 views

Algebraically, why is $\mathbb{Z}[i]/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$? [duplicate]

I understand geometrically why the Gaussian integers modulo $i+1$ is $\mathbb{Z}_{2}$, using lattices. Is there an algebraic isomorphism construction, however?
0
votes
0answers
20 views

Proof that the dual of a coalgebra is an algebra via commutative diagrams

We know that the algebra and coalgebra axioms are given via following commutative diagrams (algebra $A$ and coalgebra $C$ are over a field $\mathbb{K}$): I am now trying to show that the dual of a ...
26
votes
5answers
3k views

Commutative non Noetherian rings in which all maximal ideals are finitely generated

In commutative rings we have the following Theorem. $R$ is Noetherian if and only if each prime ideal of $R$ is finitely generated. From this Theorem I am looking for commutative rings $R$ in which ...
0
votes
1answer
48 views

Prove that (01) and (01…n-1) generate Sn? [duplicate]

Show that every element of Sn can be written as an arbitrary product of the elements (01) and (01...n-1). I understand that this can be solved using induction, and I've set up my base cases. ...
0
votes
0answers
21 views

G-set of 8 colors and 6 sides

"How many distinguishable wooden cubes can be painted if u use 8 colors (different colors on every side)" I have solved this question using Burnside's lemma ...
0
votes
1answer
61 views

Prove that $\{(1 2),(1 2 … n)\}$ can generate $S_n$ [duplicate]

I am self-studying group theory, and find it frustrating that problems on my books do not have solutions provided, here is one: Show that the symmetry group $S_n$ can be generated by the set $\{(1 ...
3
votes
2answers
50 views

Prove or disprove : the number $3+2\sqrt{-2}$ is irreducible in the ring $\mathbb{Z}[\sqrt{-2}]$

Prove or disprove: the number $3+2\sqrt{-2}$ is irreducible in the ring $\mathbb{Z}[\sqrt{-2}]$. I think it is sufficient to show that each element (except $0$) in $\mathbb{Z}\sqrt{D}$ with $D ...
1
vote
1answer
20 views

Antipode map, Hopf algebra

A Hopf algebra $H$ (over field $k$) is a bialgebra $(H,m,u,\Delta,\epsilon)$ (H, product, unit, coproduct, counit), with an antipode map, $S:H\to H$ such that $$\sum (Sh_{(1)})h_{(2)} = \epsilon(h) = ...
3
votes
1answer
50 views

Suppose A is a principal ideal domain with every ideal of finite index. Must A be a Euclidean domain?

Suppose $A$ is a principal ideal domain with every ideal of finite index (except the zero ideal). Must $A$ be a Euclidean domain? If it's not known, are there any relevant partial results?
-1
votes
0answers
30 views

Buchberger's Algorithm Example

I've been reading Ideals, Varieties and Algorithms and came across an example of Buchberger's algorithm being computed and I am not able to understand how they came to have the final result. The ...
0
votes
2answers
60 views

Prove x ⋅ e = x for all x ∈ G

Let $G$ be a finite set and $\cdot$ a binary operation on $G$ such that: The operation $\cdot$ is associative; $\forall x, y, z \in G, (x \cdot y = x \cdot z) \implies y = z$ and $(y \cdot x = z ...
1
vote
1answer
385 views

In S4, find all the even permutation and show that the set of odd permutations isn't stable for binary operations in S4.

I want to find the even permutations of $S_4$ so i am supposed to find the transpositions right? but of what permutation exactly do i find the transpositions? And how do i know which ones are even? ...
0
votes
1answer
45 views

Element order in the group

I have been reading some solution of a problem from abstract algebra and it says 'Take the maximum $n > 0$ of $\{v_2(|g|) : g \in G\}$, where G is group'. Does $v_2(|g|)$ means something? Because ...
0
votes
1answer
36 views

How to prove identity element in a set that is finite, associative, and left-right cancellable?

Let $G$ be a finite set and $*$ a binary operation on $G$ such that: The operation $*$ is associative. For all $x$, $y$, and $z$ in $G$, if $x*y=x*z$ then $y=z$ and if $y*x=z*x$ then $y=z$. I must ...
0
votes
0answers
25 views

What does the “closure of its graph” mean

I Am confused with various terminologies spelled out the same but meaning very differently depending on the situations. There are just too many. Here, I only understand the "closure" in the ...
0
votes
2answers
36 views

Showing a set is a subgroup of $S_4$

Consider a group $G=S_4$. Let H={1, (123) ,(321), (12), (13), (23)}. Show that H is a subgroup of G. What is the best way to do this. I realised <(123)>={1, (123), (321)} but I don't think this ...
0
votes
0answers
25 views

Rank of an group element

I have never heard of rank of an element $g \in G$, where $G$ is group? Does that mean order of an element or something else?
-4
votes
0answers
28 views

Prove that the intersection $K∩H$ of subgroups of group $G$ is a subgroup of $H$ [closed]

Prove that the intersection $K\cap H$ of subgroups of group $G$ is a subgroup of $H$, and that if $K$ is a normal subgroup of $G$, then $K\cap H$ is a normal subgroup of $H$.
1
vote
0answers
21 views

How many homomorphisms are there $t: \mathbb{Z}_3 \times U_8 \to S_5$?

This was a question in our quiz today and no one in class knew how to answer it correctly or are not sure). How many homomorphisms are there $t: \mathbb{Z}_3 \times U_8 \to S_5$, where $U_8$ is ...
5
votes
2answers
61 views

Salem Numbers, roots on the unit circle

There are algebraic integers which are not roots of unity , for example consider the irreducible polynomial $ P(x)= x^4-2x^3-2x+1 $. A computer software can show that this polynomial has two real ...
2
votes
2answers
89 views

Completion of the unit group of a local field

Let $K$ be a number field and $\mathfrak{p}$ a finite prime of $K$. Denote the unit group of the ring of integers of the local field $K_\mathfrak{p}$ (i.e. the completion of $K$ via $\mathfrak{p}$) by ...
1
vote
0answers
44 views

Some doubts on the relationship between Lie algebras and Lie groups

Let $(\mathbb G,*)$ be a Carnot group. Thus, by definition, $\mathbb G$ is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ admits a stratification, that is $$\mathfrak ...
0
votes
1answer
31 views

Isomorphic to Subgroup of even permutations

True or False Every finite group of odd order is isomorphic to a subgroup of $An$, the group of all even permutations. The question was in entrance exam. I think there is counter example to this ...
0
votes
1answer
42 views

Which of the following are true about the ring of continuous real valued functions C[0,1]

Let $C[0,1]$ be the space of continuous real-valued functions on the interval $[0,1]$. This is a ring under point-wise addition and multiplication. Which of the following are true: (a) For any $x ∈ ...
1
vote
4answers
40 views

Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$

Let $V$ be a vector space over a field $k$ and let $U,W$ be finite-dimensional subspaces of $V$. Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$. I know how to ...
0
votes
1answer
26 views

problems to understand a special definition of “free graded commutative algebra” from lecture

I have problems to understand a definition from lecture: Let $R$ be a commutative ring with unit and such that $2$ is invertible in $R$. The free graded commutative algebra in generators $a_1, .., ...
3
votes
1answer
17 views

Coalgebra counit property (lives in $\Bbb C\otimes C$)

I am looking at the co-unital property of a coalgebra. This is my work: Let $\epsilon: C\to \Bbb C$ be the counit, and let $\Delta$ be the coproduct of $C$, a coalgebra. So $(C,\Delta,\epsilon)$ is ...
2
votes
1answer
64 views

Possible to do well in Algebra without loving Analysis much? [closed]

Having taken some courses in higher algebra, I realized that what I truly appreciate in mathematics is abstract algebra. But it also appears that I'm not a big fan of real analysis [at least I don't ...
0
votes
2answers
44 views

Prove injectivity and sujectivity $f: ℤ$ x $ℤ$ -> {$n ∈ ℤ : 4 | n$} , $f((x,y)) = 12x - 8y$ [closed]

I am really struggling when proving injectivity and surjectivity for mult variable function. Please if someone can guide the way.
0
votes
2answers
21 views

Finding non trivial Idempotents

$A$ = the group algebra of the symmetric group $S_2$ over $\mathbb{Q}$ Find a nontrivial idempotent within $A$ So: I let $S_2 = (e,s)$. I understand that for $a \in A$ I need to find $a^2 = a$ for ...
1
vote
0answers
35 views

Splitting field of a polynomial has a solvable group of automorphisms

Prove that splitting field of a polynomial $f \in \mathbb{k}[x]$ ($f = 0$ solvable by radicals) has a solvable group of automorphisms $Aut_\mathbb{k}(\mathbb{F})$. I have just started learning Galois ...
-6
votes
1answer
47 views
1
vote
1answer
63 views

different approaches of defining tensors

This Wikipedia article says that tensor can be defined as miltilinear maps or be defined using tensor products. Could anybody explain with a simple example why these two approaches give the same ...
10
votes
2answers
845 views

Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?

For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be ...
0
votes
2answers
53 views

Exercise concerning quotient rings and ideals

I need help proving the following: Let $A$ and $B$ be rings with $\beta\in B\subseteq A$ and suppose the ideal $\beta A$ is a prime ideal. If the "/" denotes the quotient set and the equality ...
2
votes
1answer
241 views

Unique factorization domain and principal ideals .

If R was a unique factorization domain, can we deduce that for a nonzero element d in R, d has a finite number of divisors? I need this in solving this question " If R is a unique factorization ...
4
votes
1answer
168 views

Finitely generated graded modules over $K[x]$

I need some help on this exercise from A Course in Ring Theory by Donald S. Passman Find all finitely generated graded $K[x]$-modules up to abstract isomorphism. Remember, $K[x]$ is a principal ...
1
vote
0answers
16 views

irreducible polynomials over $\mathbb{F}_p[x_1,x_2]$

Recently I was reading about a irreducibility test for polynomials over the ring $\mathbb{F}_q[x]$. It is layed on the fact that the product of all monic irreducible polynomials whose degree divides ...