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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Problem regarding polynomial rings.

F is any field and let p(x) $\in$ F[x]. Also f(x) , g(x) $\in$ F[x] and deg f < deg p and deg g < deg p . We need to show that f(x) + < p(x) > = g(x) + < p(x) > implies f(x) = g(x). What ...
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1answer
28 views

Generating set of a group and even subgroup.

Let $G=\langle S\rangle$. Now let $H=\{x_1x_2\cdots x_m\mid x_i\in S \cup S^{-1},\ i\le m\in\Bbb N,\ m\text{ is even}\}$ I'm trying to prove that $[G:H]=1$ or $2$. I started doing this by proving ...
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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 ...
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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 ...
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1answer
54 views

Null Bilinear Forms $x^T A y = 0$, where $A$ is square and full rank.

Let A be a full rank square matrix (A has no null space). When does $y^T A x = 0$ occur ? It could be that this problem is case-specific, so please find attached a document where x,y, and A take ...
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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 $S$ ...
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1answer
36 views

$a^{(p-1)/n}=1$ implies $b^n=a$ for some $b\in\mathbb F_p$? [on hold]

Let $p$ be a prime. Let $n$ be a positive integer dividing $p-1$. Suppose that $a^{(p-1)/n}=1$ in the finite field $\mathbb F_p$ with exactly $p$ elements. Then does there exist $b$ with ...
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1answer
27 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 ...
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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 $p$ ...
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62 views

Order-Preserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...
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0answers
22 views

degree of generator polynomial $m(x)$

Suppose $Q$ is cyclic $(h,q)$ code over $F_u$ such that $\gcd(h,u) = 1$. Prove that degree of the generator polynomial $m(x)$ of $Q$ is $h - q$. Why do we need the condition $\gcd(h,u) = 1$ ? Any ...
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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.
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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 ...
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3answers
72 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?
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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 ...
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0answers
22 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 ...
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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. ...
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2answers
61 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 ...
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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 \} ...
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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 ...
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33 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 ...
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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 ...
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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 ...
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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?
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3answers
132 views

Only two groups of order $10$: $C_{10}$ and $D_{10}$

Show that up to isomorphism there exist only two groups of order 10: $C_{10}$ and $D_{10}$. I need some help on this question. I only know the basic definitions of isomorphism. Any hints?
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1answer
71 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 ...
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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 ...
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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 $H$.
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51 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?
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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 ...
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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) = ...
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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 ...
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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 ...
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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, .., ...
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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
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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 ...
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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.
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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 ...
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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 $\overline{\overline{L}}=\overline{L}$. ...
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2answers
33 views

Given two-sided ideals $B$ and $C$ of a ring $A$, show that $BC \subseteq B \cap C$

Given two-sided ideals $B$ and $C$ of a ring $A$, (a) show that $BC \subseteq B \cap C$. (b) If the ring $A$ is commutative and $B + C = A$, show that $BC = B \cap C$. Here's what i have but I am ...
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4answers
41 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 ...
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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 ...
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1answer
47 views

The sum of invertible matrices is also invertible? [closed]

The sum of invertible matrices is also invertible ? THANKKSS!!!
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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 ...
3
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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 ...
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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 ...
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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 ...
1
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1answer
22 views

An isomorphism between product of number fields, contains the same number of factors

Suppose that we have an isomorphism of rings $$f:K_1\oplus\cdots\oplus K_r\to K'_1\oplus\cdots\oplus K'_s,$$ with $K_i$'s and $K_j'$'s are a number fields, the sum and the product are componentwise. ...
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0answers
35 views

$S_X = \{f(x) = x : \text{ bijective} \}$. prove $S_x$ is isomorphic to $S_n$

Aright, to start $S_n$ is the Symmetric group and $S_X = \{x_1, x_2, \ldots x_n\}$. Going through the mapping $\phi(S_X) \to S_n$, I'm not sure how I'd show this mapping and the first thought that ...
1
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1answer
42 views

Find the Galois group of the field $\mathbb{k}_{sym}(x_1,\dots,x_n)(D)$

Find the Galois group of the field $\mathbb{k}_{sym}(x_1,\dots,x_n)(D)$ over the field $\mathbb{k}_{sym}(x_1,\dots,x_n)$, the characteristic of $\mathbb{k}$ is 2 and $$D(x_1,\dots,x_n) = \prod_{1 \leq ...