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 the subset $\overline{I} = \{\overline{x}:x \in I\}$ is an ideal.

Assume that $I$ is an ideal of the ring $\mathcal{O}_d = \left\{ \begin{array}{ll} \mathbb{Z} [\sqrt{d}] & \text{ if } d \text{ is even } \\ \mathbb{Z} [ \frac{1 + ...
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3answers
23 views

Show that if $A, B$ are groups, then $A \times B$ is solvable if and only if both $A$ and $B$ are solvable?

Show that if $A, B$ are groups, then $A \times B$ is solvable if and only if both $A$ and $B$ are solvable? Why is obvious that if $A$ and $B$ are solvable then $A \times B$ is solvable?
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2answers
53 views

Proof that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$

I tried to prove one of the examples in my Abstract Algebra book that stated: Prove that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$ I went about just saying that $a^4b = ba ...
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2answers
23 views

Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ then $Z(G)$ is a group

So my challenge is: Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ where $G$ is a group, then $Z(G)$ is a group For the identity, $e$ clearly is in $Z(G)$ and in ...
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3answers
31 views

Show that $f: \mathbb{Z} \to n\mathbb{Z}$ defined as $f(m) = mn$ is bijective

Let $n \in \mathbb{N}$. Show that $f: \mathbb{Z} \to n\mathbb{Z}$ defined as $f(m) = mn$ is bijective such that $f(m_1 + m_2) = f(m_1) + f(m_2)$, $\forall m_1, m_2 \in \mathbb{Z}$ To be bijective, ...
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6answers
84 views

Show that $x^{3}-3$ irreducible over $\mathbb{Q}(\sqrt{-3})$

Is there a slick way to show that $x^{3}-3$ is irreducible over $F= \mathbb{Q}(\sqrt{-3})$? What I did seems kind of convoluted (showing directly that there is no root in F). Thanks
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1answer
18 views

algebraic integers of $\mathbb{Q}(\sqrt{d})$

Assume that $d$ is square-free. What is the set of algebraic integers in $\mathbb{Q} \left(\sqrt{d} \right) = \{a + b \sqrt{d}:a,b \in \mathbb{Q} \}$? The algebraic integers in ...
4
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1answer
22 views
0
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6answers
29 views

$\gcd(a,n)\neq 1 \implies $ there is $b$ such that $ab\equiv 0 \pmod{n}$

I have that $\gcd(a,n)\neq 1$ ($a$ and $n$ are not coprime). Then, somehow, I need to prove that exists an $b$ such that $$ab\equiv 0 \pmod{n}$$ What I did: $$ab\equiv 0 \pmod{n}$$ is the same ...
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1answer
26 views

Why does it mean that $n$-th variable is removable?

I'm reading the proof for "the fundamental theorem of symmetric polynomials" and I have a trouble with it (http://en.m.wikipedia.org/wiki/Elementary_symmetric_polynomial) Let $P(X_1,...,X_n)$ be a ...
5
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1answer
51 views

Extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, splitting.

In the extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, must principal prime ideals of $\mathbb{Z}[\sqrt{-5}]$ necessarily split into 2? Must nonprincipal prime ideals not split? ...
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2answers
44 views

Algebra and Maximal ideal.

I am trying to solve the following problem. If $ \mathcal{K}$ is a field and $a_1,a_2,\dots,a_n \in \mathcal{K}$. Prove that $(x_1-a_1,x_2-a_2,\dots,x_n-a_n)$ is a maximal ideal in ...
3
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1answer
49 views

Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent

can you find a Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent?
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0answers
23 views

Fundamental theorem of Algebra using ideas of complex singularities

Below is an excerpt from Arnold's Theory of Catastrophes (I haven't got an American edition, so translating from Russian). Where I can read about it in more detail? Not only regarding polynomials. ...
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0answers
19 views

Looking for a non trivial homomorphism II [on hold]

Is there a non trivial homomorphism $f: SU(2) \to Diff(S^1)$?
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1answer
56 views

Looking for a non trivial homomorphism I

Is there a non trivial homomorphism $f: SU(2) \to O(2)$? Is there a concrete description of $Hom(SU(2), O(2))$?
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0answers
15 views

Is $\alpha$ a norm in the extension $K(\sqrt[n]{\alpha})$?

I'm having trouble wrapping my head around this. $K$ is a field of characteristic zero containing all $n$th roots of unity, and $\alpha \in K$. Let $L = K(\sqrt[n]{\alpha})$, $\mu$ the minimal ...
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0answers
10 views

Why is an $n$th power a norm in a Kummer extension?

Let $F$ be a $p$-adic field containing the $n$th roots of unity. Then by Kummer theory, $[F^{\ast} : F^{\ast n}]$ (which is finite) is equal to the cardinality of $\textrm{Gal}(E/F)$, where $E$ is ...
0
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1answer
13 views

For any monomial ordering, $1\leq m$ for any monomial $m$

Let $R$ be a ring. Let $\leq$ be a well-ordering on the set of (monic) monomials in $R[X_1,...,X_n]$. Then, $\leq$ is said to be a monomial ordering iff $mm_1\leq mm_2$ whenever $m_1\leq ...
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2answers
35 views

A normal subgroup so that any homomorphism into a $p$-group is trivial on it.

Problem Let G be a finite group of order $n$ and $p|n$. Show that there is a unique normal subgroup $N$ satisfying the following property: (1)$G/N$ is a $p$-group (I guess it can be trivial group). ...
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3answers
53 views

A finite group which has a unique subgroup of order $d$ for each $d\mid n$.

Problem Suppose G is a finite group of order $n$ which has a unique subgroup of order $d$ for each $d\mid n$. Prove that $G$ must be a cyclic group. My idea: I try to prove it by induction. Let ...
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1answer
47 views

How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
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1answer
20 views

Show that a representation of a finite group is isomorphic to its dual if its character takes only real values

This appeared as a part of showing that a representation of a finite group is isomorphic to its dual if and only if its character takes only real values. The "only if" part was easy to show. For the ...
3
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2answers
40 views

Algebra generated by a single element over an infinite field, does $A_1 \times \cdots \times A_r$ has the same property?

Let $K$ be an infinite field and $A_1, \ldots, A_r$ algebras over $K$ finite dimensional and such $\forall i = 1, \ldots, r \ \exists x \in A_i : A_i = K[x]$ (I think we say that $A$ is finitely ...
2
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1answer
64 views

Matrix and field extension

It is given that $F\subset K$ are fields. $A$ is a matrix of size $n\times n$ over $K$. I need to prove that there exist $c_1,\ldots,c_k\in K$, linearly independent over $F$, and matrices ...
1
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1answer
35 views

Proof that if $H_1 \leq G$ and $H_2 \leq G$ then $H_1 \cap H_2 \leq G$

I am trying to prove that Prove that if $H_1 \leq G$ and $H_2 \leq G$ then $H_1 \cap H_2 \leq G$ Unlike this question: Prove $H_1 \cap H_2 \le H_1 $ when $H_1, H_2 \le G$ and $H_1$, $H_2$ are ...
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1answer
50 views

What is orthogonal group $O(1)$?

I know that $O(2)$ is the group of 2x2 orthogonal matrices, but how can we extend the meaning of group and orthogonal to $O(1)$?
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40 views

Show that if for $a \in \mathbb{Q}$ with $0 = f(a) $ for a monic polynomial $f(x)\in \mathbb{Z}[x] $, then $a \in \mathbb{Z}$

I. Show that if for $a \in \mathbb{Q}$ with $0 = f(a) \in \mathbb{Z}[x] $ for a monic polynomial, then in fact $a \in \mathbb{Z}.$ II. The set of algebraic integers of $\mathbb{Q}$ is ...
3
votes
2answers
65 views

Is the left translation $T_a(x) =ax $ a homomorphism?

I apologize if this is a super basic question but I was reading Lang's undergraduate algebra book and it says that the following function is a homomorphism: $$T_a(x) = ax$$ The way I would check if ...
0
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1answer
41 views

Looking for a direct proof that all maximal ideals of $\mathbb C[x_1,x_2,…,x_n]$ are generated by $n$ linear polynomials

Without using Hilbert's Nullstelensatz , can we directly prove that all maximal ideals of $\mathbb C[x_1,x_2,...,x_n]$ is of the form $\langle x-a_1,x-a_2,...,x-a_n \rangle$ ? It is easy to prove it ...
0
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4answers
52 views

How do we find the minimal polynomial of $\alpha = a + b \sqrt{d}$ over $\mathbb{Q}$?

Let $\alpha = a + b \sqrt{d} \in \mathbb{Q} \left(\sqrt{d} \right) = \{a+b \sqrt{d}:a,b \in \mathbb{Q} \}.$ The minimal polynomial $m(x)$ of an algebraic number $\alpha \in \mathbb{C}$ is the monic ...
0
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1answer
42 views

Is there a function over $\mathbb{Z}_p$ that is never linear?

Let $p$ be a prime. I wonder if there is a function $f$ that satisfies the following rule: Whenever $$z_1 + \dots + z_c \equiv cx \mod p$$ (where $1 < c < p$, and $z_j, x \in \mathbb{Z}_p$) ...
2
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1answer
45 views

Any curve of genus three is either hyperelliptic or trigonal?

A curve $C$ is said to be trigonal if it admits a rational function $f: C \to \mathbb{CP}^1$ of degree $3$. Is any nonsingular plane curve of degree four trigonal? Can the map be chosen so that it has ...
0
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1answer
69 views

Prove the isomorphism of categories $Fun(\mathcal{A}\times\mathcal{B},\mathcal{C})\cong Fun(\mathcal{A},Fun(\mathcal{B},\mathcal{C})),$

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
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2answers
24 views

Show that if $\phi(F) \neq \{0\}$ then $F \cong R$.

Let $F$ be a field. Let $R$ be a ring and suppose $\phi : F \rightarrow R$ is an onto ring homomorphism. Show that if $\phi(F) \neq \{0\}$ then $F \cong R$. (Prove $F$ isomorphic to $F/\{0\}$ first) ...
0
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1answer
23 views

Proving two matrices are cogredient over $\mathbb{Q}$

Two matrices $A,B$ are said to be cogredient if there exists an invertible matrix $P$ such that $B = P^{t}AP$. I know how to tell if two matrices are cogredient in algebraically closed fields, its as ...
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0answers
34 views

If $G$ and $H$ are groups, prove that $(G \times H, x)$ is a group.

Prove that, if $(G,\ast)$ and $(H,\bullet)$ are groups, then the Cartesian Product $G \times H$ with the operation $(g_1,h_1) \circ (g_2, h_2) := (g_1 \ast g_2, h_1 \bullet h_2)$ $(G \times H, ...
2
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1answer
48 views

Divisors of differentials.

Let $\mathbb{C}$ be the base field. Suppose $C \subset \mathbb{P}_2$ is a nonsingular projective curve of degree $d > 3$. Must it be the case that $D \in \text{Div}(C)$ is the divisor of a ...
2
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1answer
30 views

Existence of multiplicative inverse in field $\mathbb{Q}(\sqrt{d})$

Let $\mathbb{Q}(\sqrt{d}) = \{a + b \sqrt{d}: a,b \in \mathbb{Q} \}.$ Show that $\mathbb{Q}(\sqrt{d})$ is a field. Everything seems obvious except for existence of inverses in the multiplicative ...
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0answers
18 views

Verify regular representation?

Let $G$ be a finite group and let $V$ be the vector space of functions from $G$ to $\mathbb{C}$. For $g \in G$ and $f \in V$, let $R(g)(f)$ be the function $$(R(g)f)(x) = f(xg^{-1}).$$ How can I show ...
3
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2answers
43 views

How to define a specific ring using a homomorphism

If we have a ring $R$ then I can form a ring of matrices isomorphic to $R$ by setting $r \overset{\phi}{\mapsto} \left( \begin{array}{ccc} r & 0 \\ 0 & 0 \end{array} \right) $ and defining ...
8
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2answers
58 views

What is $\mathbb{Z}[i] \otimes_{\mathbb{Z}[2i]} \mathbb{Z}[i]$?

What is $\mathbb{Z}[i] \otimes_{\mathbb{Z}[2i]} \mathbb{Z}[i]$? Also, since $\mathbb{Z}[i]$ is a PID, we should be able to write this $\mathbb{Z}[i]$-module as a direct sum of cyclic ...
9
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0answers
41 views

$M_1 \to M_2 \to M_3 \to 0$ exact iff $0 \to \text{Hom}_A(M_3, N) \to \text{Hom}_A(M_2, N) \to \text{Hom}_A(M_1, N)$ is exact.

Let $M_1 \to M_2 \to M_3 \to 0$ be a sequence of homomorphisms of $A$-modules. Is this sequence exact if and only if the induced sequence of abelian groups$$0 \to \text{Hom}_A(M_3, N) \to ...
2
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1answer
29 views

real valued functions with composition

If $G$ is the set of all functions $f : \mathbb{R} \to \mathbb{R}$ such that $f(x) \ge 0$ for all $x \in \mathbb{R}$ with $f ∗ g = f \circ g$ (here $\circ$ denotes the operation of composition), for ...
2
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2answers
41 views

Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$

My challenge is Prove that if $n = 0,1,2,\ldots$ and $n\Bbb Z = \lbrace nk: k \in \Bbb Z \rbrace$, show that $n\Bbb Z$ is a subgroup of $\Bbb Z$ and are the only subgroups. I handled the first ...
7
votes
2answers
90 views

For any rng $R$, can we attach a unity?

Let $R$ be an rng. (There may be no unity) Then, does there always exist a ring(with unity) $A$ such that $R$ is a subrng of $A$?
0
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1answer
31 views

I have to show center of $M_n(H)$ is $\mathbb{R}I$ [on hold]

Let $H$ be a real quaternion ring. I have to show that the center of $M_n(H)$ is $\mathbb{R}I$. Can anyone help?
1
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2answers
14 views

Proof of the right and left cancellation laws for Groups

I was asked to proof the right and left cancellation laws for groups, i.e. If $a,b,c \in G$ where $G$ is a group, show that $ba = ca \implies b=c $ and $ab = ac \implies b = c$ For the first ...
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1answer
17 views

Concavity and quasiconcavity… [on hold]

How do you explain the difference between concavity and quasi concavity? or convexity and quasi convexity?
3
votes
2answers
36 views

Fiber product of groups

Let $f: G \longrightarrow K$, $g: H \longrightarrow K$ be group homomorphisms. For the fiber product $G \times_K H$ to exist, is it necessary to require that $f$ and $g$ be surjective? I can't see ...