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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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
18 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 ...
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2answers
31 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 ...
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1answer
59 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 ...
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1answer
30 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
39 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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37 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 ...
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1answer
37 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 ...
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1answer
38 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 ...
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31 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 ...
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1answer
38 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 $z_j, x \in \mathbb{Z}_p$) ...
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1answer
41 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 ...
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1answer
53 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
19 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) ...
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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
32 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, ...
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1answer
39 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 ...
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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
14 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 ...
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40 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 ...
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2answers
54 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
35 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
26 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
40 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 ...
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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 $A$?
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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?
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2answers
13 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?
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34 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 ...
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1answer
12 views

Possible angles between roots in a root system

Given a Root System $\Phi$ let $\alpha,\beta \in \Phi$ with $\alpha \neq \pm \beta$ and $||\beta||\geq ||\alpha||$. Let $\theta$ be the angle between $\alpha$ and $\beta$. Since $<\alpha,\beta> ...
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33 views

Is the subgroup of a non-abelian group is non-abelian?

Is the following statement always true Subgroup of a non-abelian group is non-abelian
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26 views

Endomorphism commutes with its adjugate

Let $R$ be a commutative ring, $M$ a free $R$-module of rank $n$ and $f \in \rm{End}(M)$. The adjugate $f^\sharp$ of $f$ is defined by the equalities $$ f^\sharp(x) \wedge y = x \wedge ...
3
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2answers
49 views

Proving that disjoint unions of presentations are coproducts of groups

I'm working through Aluffi's Algrebra: Chapter 0 and I need some assistance with an excercise. Aluffi, Ex. II.8.7 Let $(A|R)$, resp. $(A'|R')$, be a presentation for a group $G$ in Grp, resp. ...
2
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1answer
24 views

$\operatorname{Rad}(k)=\operatorname{Rad}(L)$

Given a Lie Algebra $L$ on a field $F$, we define the radical of $L$ $\operatorname{Rad}(L)$ as the largest solvable ideal of $L$. We define the adjoint representation ...
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1answer
45 views

Classifying groups of order 18

I am trying to classify groups of order 18. So far, I have shown that a group $G$ of order 18 is given by $G\cong C_9 \rtimes_{\varphi} C_2$ or $G\cong (C_3 \times C_3)\rtimes_{\varphi} C_2$. If ...
5
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4answers
56 views

Prove that $f=x^6+ax+5$ is reducible over $\mathbb{Z_7},\forall a\in\mathbb{Z_7}$

We have $f=x^6+ax+5\in\mathbb{Z_7}$ and we have to show that it is reducible on $\mathbb{Z_7}$, $\forall a\in\mathbb{Z_7}$. Here are all my steps: For $a=0$ we'll get $f=x^6+5\in\mathbb{Z_7}$. But ...
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1answer
41 views

Fraction modulo integer in sage [on hold]

I'm working on a sage script right now, I have some polynomials coefficients that are rational, and I want to apply a congruence on these coefficientss, for example: $p = 1 + (7/2)x$ the function ...
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26 views

lattices and torsion free sublatices.

I have the following statement that I cant proof, which according to my book is trivial. Let $N$ be a lattice. Let $N_1 \subset N$ be a sublattice such that $N/N_1$ is torsion free. Then it followes ...
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1answer
14 views

$UTM_n[D]$ is artinian

Why is the upper triangular matrices over a division ring D is artinian? I tried to find properties of this class of rings. The only thing I found that the jacobson radical of this ring is the ...
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1answer
50 views

To find the no. of elements of order $7$ in a field of 8 elements

Let $F$ be a field of $8$ elements and $A= \{x\in F \,|\, x^7=1 \text{ and } x^k\neq 1 \text{ for all natural number $k<7$}\}$. Then the number of elements in $A$ is: a) 1 b) 2 c) 3 d) 6 ...
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1answer
32 views

Free module over a set

I think I understand the definition of a free $R$-module over a set $X$: it is given by the set of all maps from $X$ to $R$ which vanish at all but finitely many points of $X$. The module operations ...
4
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2answers
56 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the $2 \times 2$ matrix ring $M_2(\mathbb{C})$. let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional ...
3
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1answer
24 views

The coproduct of a family of objects of a Preorder (seen as a category)

If the coproduct of a family of objects of a Poset (seen as a category) is the least upper bound, who is the coproduct of a family of objects of a Preorder (seen as a category)? My intuition ...
2
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1answer
21 views

Perfect pairing induces isomorphism of tensor products

Let $M, N$ be $R$-modules and $(\cdot, \cdot): M \times N \to R$ be a perfect pairing. Wikipedia sais that this means that the map $\varphi: M \to \text{Hom}_R(N, R), m \mapsto (n \mapsto (m, n))$ is ...
2
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2answers
22 views

Cyclic projective module

Let $R$ be an integral domain. If $M$ is a cyclic $R$-module which is also projective, then must there necessarily be an isomorphism of $R$-modules $M \cong R$?
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1answer
34 views

Relation of $\operatorname{Ext}$ and projective dimension

I have some problem to understand the proof of proposition 8.38, page 473 from An Introduction to Homological Algebra by Rotman. Proposition: Let $x\in Z(R)$ be an element which is not a zero-divisor ...
5
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4answers
162 views

Showing that $7+\sqrt[3]{2}$ is an algebraic number

How do I go about showing that $7+\sqrt[3]{2}$ is an algebraic number? I need to show that it is the root of an integer valued formal polynomial? How do I solve these problems in general? I haven't a ...
0
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1answer
21 views

Prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective

In the situation $(_RA,_RC_S)$, prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective. I appreciate your help.
0
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0answers
26 views

Is $\oplus_{i \in I} M_i$ necessarily projective?

If $\{M_i\}$ is a collection of projective modules over a ring $R$, then must the direct sum $\oplus_{i \in I} M_i$ necessarily be projective? I understand that each direct sum of two modules is ...
2
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1answer
24 views

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module?

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module? I understand I am supposed to think of $A$ as an $A \times B$-module by identifying ...