0
votes
2answers
69 views

Fields - Proof that every multiple of zero equals zero

This is one of my first proofs about fields. Please feed back and criticise in every way (including style and details). Let $(F, +, \cdot)$ be a field. Non-trivially, $\textit{associativity}$ implies ...
3
votes
1answer
128 views

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
3
votes
1answer
45 views

Groups - Prove that every element equals inverse of inverse of element

This is my first proof about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. We use $^{-1}$ to denote inverse ...
0
votes
0answers
49 views

Prove that $T^{4} -12T^{2} +64$ is irreducible in $\mathbb{Q}[T]$

Is the following correct? I choose $3$ irreducible in $\mathbb{Z}$. If $g=(1+(3))T^{4} - (12 + (3))T^{2} + (64 + (3)) \in \mathbb{Z[T]}/(3)$ is irreducible, then $f=T^{4} -12T^{2} +64 \in ...
1
vote
1answer
49 views

Exercise on characterization of free abelian groups

I was wondering if someone can please check my work on a homework problem. This is from the graduate Hungerford text. Chapter 2.1, number 3. Let $X=\{a_i\ |\ i\in I\}$ be a set. Then the free abelian ...
0
votes
2answers
44 views

Proving that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$

Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ and let $a,b \in E$ where $E$ is some extension of $F$. If $a$ is a zero of $f(x)$ and $b$ is a zero of $g(x)$, show that $f(x)$ is ...
1
vote
0answers
49 views

Deduction of usual Cayley-Hamilton Theorem from “Determinant Trick”

Here is a statement of a standard theorem in commutative algebra (see page 60 of this book): Theorem. ("Determinant Trick") Suppose that $R$ is a commutative ring with $1$. Let $M$ be a finitely ...
1
vote
1answer
67 views

Using resultants to check if multivariate polynomials have a common factor - is my proof correct?

Proposition: Let $f, g \in \mathbb R[x,y,z]$. Then the condition that $f, g$ have a common polynomial factor is an algebraic condition on their coefficients. By algebraic condition, I mean there is a ...
2
votes
0answers
37 views

Kernels of induced maps in cohomology

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ ...
3
votes
2answers
50 views

Group theory exercise - verification?

I'm self-studying abstract algebra, and this is the first non-trivial group theory exercise I've done. Although it's a well-known result, I'd like to make sure it is correct as it took a good few ...
1
vote
2answers
83 views

Show that $Z\times Z$ is not cyclic… [duplicate]

The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra. Recall that an infinite cyclic group is isomorphic to ...
1
vote
0answers
50 views

Direct proof that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p^2\mathbb{Z}$-module.

I am trying to prove that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p\mathbb{Z}$-module. The reasoning I have is the following. We have an exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} ...
0
votes
2answers
42 views

Prove that the order of $U(n)$ is even when $n>2$.

I'm trying to provide a solution to the following claim: "The order of $U(n)$ is even when $n>2$." Note: here, $U(n)$ is the set of all positive elements that are less than and relatively prime ...
2
votes
1answer
38 views

Please check my proof on: $\sim$ is an equivalence relation $\Leftrightarrow S<G$

Problem: Let $\emptyset\ne S\subset G$, where $G$ is a group, and define a relation on $G$ by $a\sim b\Leftrightarrow ab^{-1}\in S$. Show that $\sim$ is an equivalence relation if and only if $S$ is a ...
1
vote
2answers
47 views

Prove that the field of quotients of an integral domain $D$ is the smallest field containing $D$. . My Attempt Shown

Let $D$ be an integral domain and let $F$ be the field of quotients of $D$. Show that if $E$ is any field that contains $D$, then, $E$ contains a sub field that is ring isomorphic to $F$. Hence, the ...
0
votes
1answer
33 views

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$. Show that the mapping $x \rightarrow ax$ is a ring homomorphism

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$. Show that the mapping $x \rightarrow ax$ is a ring homomorphism Attempt: Let $\Phi: Z_m \rightarrow Z_n$ be a ring homomorphism ...
1
vote
2answers
36 views

Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$. My Attempt Shown

Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$. Attempt: Let $F'$ be the field of Quotients of the field $F$. Let $\Phi:F \rightarrow F'$ such that ...
0
votes
0answers
16 views

Prove that both $R_p$ and $R^p$ are abelian groups under ordinary addition of rationals.

Q: Let $p$ be a fixed prime. Let $R_p$ be the set of all those rational numbers whose denominator is relatively prime to $p$. Let $R^p$ be the set of rationals whose denominator is a power of $p\ ...
1
vote
1answer
38 views

Proving the Well-Ordering Property

My textbook states the Well-Ordering property as following: If $A$ is any nonempty subset of $\mathbb{Z}^+$, there is some element $m \in A$ such that $m \le a$, for all $a$ in $A$ ($m$ is called ...
1
vote
0answers
27 views

The only fixed-point free automorphism of order $2$ is $\phi(a)=a^{-1}$(in a finite group)

I got the problem in Dummit and Foote's Algebra book to prove if $G$ is a finite group that has an automorphism $\phi$ in which if $a=\phi(a)$ then $a=1$. And which satisfies $\phi(\phi(a))=a$ for ...
7
votes
4answers
94 views

Herstein Question: $G^{i}$ normal in $G$?

I just wanted to ask a quick question. I'm going over the second edition of I.N. Herstein's topics in algebra and one of his exercises asks the reader to prove that each $G^{i} $ is a normal subgroup ...
1
vote
1answer
16 views

Determining if any of these three are an ideal of $\mathbb{R}[x]$

$\mathbb{R}[x]$ denotes the ring of polynomials in $x$ with real coefficients. Let $I \subset \mathbb{R}[x]$ be the subset of those polynomials with constant coefficient $0$, and let $J \subset ...
1
vote
1answer
39 views

Proof of coset and normal subgroup

I have this question: Let $G$ be a group, $a,b\in G$ and let $H$ be a subgroup of $G$. i) Give the definition of the coset $aH$ ii) Prove that $aH = bH$ if and only if $a^{-1}b\in H$ ...
0
votes
1answer
34 views

Proving $M_p$ is maximal in $C[0,1]$

Let $M_p$ be the ideal of those continuous functions of $C[0,1]$ which have $p\in [0,1]$ as a zero. It is a commonly known fact that $M_p$ is a maximal ideal. However, the proof is generally ...
2
votes
0answers
39 views

Counter example of monotone union [duplicate]

I saw this exercise in "Elements of Abstract and Linear Algebra" by E. H. Connell: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, ...
2
votes
3answers
70 views

Monotone Union of subgroups being subgroup

I saw this exercise in a book: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, if $\{H_t\}$ is a monotonic collection, show that ...
1
vote
1answer
29 views

Algebra subgroup question

Let $G$ be a group, and let $H$ be a subgroup of $G$. Define $$C_G(H) := \lbrace g \in G \mid h \in H :gh=hg \rbrace.$$ (The set $C_G(G)$ is called the centralizer of $H$ in $G$.) Show that $C_G(H)$ ...
1
vote
1answer
53 views

Proof Check: automorphism sends primitive root to primitive root

I was just wondering if this is a valid proof. I am assuming knowledge that if $\phi$ is an automorphism of a numeric field the $\phi$ fixes $\mathbb{Q}$. Also, if $\phi \in$ ...
2
votes
1answer
47 views

If a group doesn't have subgroups of index 2 and 3, then any subgroup of index 4 is normal.

Let $G$ be this group and $H$ be any subgroup of index 4. $G$ acts on the set of left cosets of $H$ in $G$, which is a homomorphism $\varphi: G\to Aut(G/H) = S_4$. It is easy to see that $\ker ...
1
vote
0answers
24 views

Subgroups of SO$(2)$

I'm doing an independent study with John Stillwell's Naive Lie Theory and I wanted to know if I'm on the right track. I'm just looking for some confirmation that these are acceptable answers. Find ...
3
votes
2answers
29 views

Zero divisors in ring of real valued functions.

I'm working though Pinter's A book of Abstract Algebra and would like a quick verification on a simple problem. Exercise 17.B2 asks Describe the divisors of zero in $\mathcal{F}(\mathbb{R})$. ...
2
votes
3answers
255 views

Is this example right (ideals of $\mathbb{Z}[x]$)?

I encountered the following problem: Let $I_{0}=\{f(x)\in \mathbb{Z}[x] \ | \ f(0)=0\}$. For any positive integer, show that there exists a sequence of ideals such that $I_0\subsetneq ...
1
vote
1answer
38 views

Prove: $(\emptyset,\{\emptyset\})$ is an algebra of sets

I must prove the following: Prop.: $(\emptyset,\{\emptyset\})$ is algebra of sets Proof: $\emptyset \in \{\emptyset\} $ by hypothesis $\emptyset -\emptyset=\emptyset$ and by hypothesis $ ...
2
votes
0answers
56 views

Is my proof correct? (About $SL(2,\Bbb Z)$)

Problem: Let $A=\begin{bmatrix}0 &-1\\1 &0\end{bmatrix}, B=\begin{bmatrix}0 &1\\-1 &1\end{bmatrix}\in GL(2,\Bbb Q)$, let $N=\langle \pm I\rangle$ and let $M=\langle ...
1
vote
1answer
42 views

Orbit-Stabiliser Theorem Application

Question Let $G$ be the symmetric group $S_n$ acting on the $n$ points $\lbrace 1, 2, 3, . . . , n\rbrace$, let $g\in S_n$ be the n-cycle $(1,2,3,. . . , n)$. By applying the Orbit-Stabiliser Theorem ...
12
votes
0answers
107 views

Let $f$ and $g$ be two group homomorphisms from $G$ to $G'$. Is $H$ a subgroup of $G$?

Can someone please verify this? Let $f$ and $g$ be two group homomorphisms from $G$ to $G'$. Let $H \subset G$ be the subset $\{x \in G: f(x)=g(x)\}$. Is $H$ a subgroup of $G$? Let $e$ and $e'$ ...
2
votes
0answers
45 views

Prove that the center of a group is normal.

Can someone please verify this? Prove that the center of a group is normal. let $G$ be a group, and let $\operatorname{Z}(G)$ denote its center. Let $g \in G$ and $z \in \operatorname{Z}(G)$. ...
2
votes
1answer
39 views

Show that $f:\mathbb{R}^+ \longrightarrow \mathbb{C}^\times$ defined by $f(x)=e^{ix}$ is a homomorphism

Can someone please verify my proof? Show that $f:\mathbb{R}^+ \longrightarrow \mathbb{C}^\times$ defined by $f(x)=e^{ix}$ is a homomorphism, and determine its kernel and image. Let $x$ and $y$ ...
2
votes
1answer
56 views

Which of the following groups are subgroups?

I've written an answer for an exercise in Artin's algebra. Can someone please verify it? Which of the following groups are subgroups? (a) $GL_n(\mathbb{R}) \subset GL_n(\mathbb{C})$ (b) ...
0
votes
0answers
28 views

Artin algebra chapter 2.3 exercises

I've written some proofs from Artin's book. Can someone please verify them? Prove that the additive group $\mathbb{R}^+$ of real numbers is isomorphic to the multiplicative group $P$ of positive ...
3
votes
0answers
50 views

Artin chapter 2 exercises

I've written some solutions to Artin's exercises. Can someone please verify them? Assume the equation $xyz = 1$ holds for a group $G$. Does it follow that $yzx=1$? That $yxz=1$? We first show ...
4
votes
5answers
332 views

Prove any subgroup of a cyclic group is cyclic.

was just wondering if this is a valid proof for the aforementioned question? I am quite confident that it isn't, but not exactly sure why. Maybe I am missing the point of proofs by induction ...
8
votes
3answers
174 views

Polynomial irreducible - maximal ideal

I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal. $I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$ $\mathbf 1)$ Is $I_1$ a maximal ideal in ...
2
votes
3answers
136 views

Group Homomorphism Questions (my attempts shown)

(a) Let $p$ be a prime. Determine the number of homomorphisms from $\Bbb Z_p \oplus \Bbb Z_p$ into $\Bbb Z_p$. Attempt: Suppose $\Psi:Z_p \oplus Z_p \rightarrow Z_p$ is an into homomorphism. ...
0
votes
0answers
55 views

Is my solution correct? Finite abelian groups are CLT groups.

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: Use Cauchy's Theorem and induction to show that a finite abelian group has a subgroup of order $n$ for each ...
4
votes
1answer
42 views

Dual of Schanuel lemma

This is an exercise from Rotman, Introduction to homological algebra. Given exact sequences of $R$-modules \begin{array}{ccccccccc} 0 & \longrightarrow & M & \overset{i}{\longrightarrow} ...
1
vote
1answer
46 views

Confused about a particular example of rational canonical form… please help me find my error.

The minimal polynomial of the matrix $A = (\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix})$ is $x^2 + 1$. (At least, I think so - how can one be sure about this?) If we think of this ...
0
votes
0answers
11 views

Verification of step in a proof of the decomposition of primary f.g torsion modules over PIDs?

I was reading about the decomposition of finitely generated primary torsion modules over PIDs, and though of an alternative way to do the "inductive" step. Since it is substantially simpler than the ...
1
vote
1answer
36 views

$\hom_{\mathbb{Z}}(\mathbb{Z}_n, G) \cong G[n]$

I'm doing this exercise: G is an abelian group, prove that $$\hom_{\mathbb{Z}}(\mathbb{Z}_n, G) \cong G[n]= \lbrace g \in G | ng = 0 \rbrace$$ My attempt: Let's consider the exact sequence $$ 0 \to ...
0
votes
0answers
51 views

Would like to confirm answer (regarding sets)

As you might know from my precious questions, I am pretty weak with quantifiers. Below is my solution to the stated problem, if incorrect, could someone explain why? My attempted solution: ...