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)

1
vote
1answer
40 views

Determine if the cyclic group $\langle(1234)\rangle$ is normal in $S_4$ [duplicate]

How exactly can I determine if it is normal or not?
1
vote
1answer
164 views

Proof that field of constants is trivial

Let $d$ be $k$-derivation of $L=k(x_1,x_2,\dots)$ (field of rational functions over field k) defined by $d(x_i)=x_{i+1}$ for $i=1,2,\dots$. Then $L^d=k$. Book says it's easy to prove it but i dont ...
1
vote
1answer
73 views

Show that A is an algebra

Suppose $X$ is a collection of sets and $Ω$ element of $X$. Also $A$, $B$ are elements of $X$. Then, $A-B=A\cap B^c$ element in $X$. Show that $X$ is an algebra. Please help me, how I can show this ...
0
votes
3answers
106 views

Finding units in cyclotomic fields

I want to classify the six units in $\mathbb Z[\zeta_3]$, where $\zeta_3$ is a primitive cube root of unity. I know the basic idea of this is to show that the norm of $\alpha \in \mathbb Z[\zeta_3]$ ...
3
votes
1answer
47 views

One-sided nilpotent ideal not in the Jacobson radical?

Problem XVII.5a of Lang's Algebra, revised 3rd edition, is: Suppose $N$ is a two-sided nilpotent ideal of a ring $R$. Show that $N$ is contained in the Jacobson radical $J: = \{ \cap\, I: I ...
4
votes
1answer
94 views

Show that $\left| N \cap Z(G) \right| > 1$

Let $G$ be a non-trivial $p$-group, let $Z(G)$ denote its center, and let $N$ be a non-trivial normal subgroup of $G$. Show that $\left| N \cap Z(G) \right| > 1.$ Proof. Let $a \ast n = C_a(n) ...
1
vote
4answers
52 views

Show that there exists a polynomial $p(X) ∈ F[X]$ satisfying $c^n + c^{−n} = p(c + c^{−1})$.

Let $c$ be a nonzero element of a field $F$ and let $n > 1$ be an integer. Show that there exists a polynomial $p(X) ∈ F[X]$ satisfying $c^n + c^{−n} = p(c + c^{−1})$. I made some particular ...
2
votes
1answer
59 views

Powers in non-commutative rings

Let $a,b$ be elements of a non-commutative ring $R$ with $\operatorname{char}(R) =p > 0$ and suppose that $ab-ba=[a,b]=1$. My question is simply: Could you give a formula for the element $(a^n ...
1
vote
1answer
90 views

Is the matrix of the sum of two linear maps equal to the sum of the matrices of the two maps?

Is the matrix of the sum of two linear maps equal to the sum of the matrices of the two maps? We have bases $\{v_1,\dots,v_n\}$ of $V$ and $\{w_1,\dots,w_m\}$ of $W$. $M(T+S) = M(T) + M(S)$? ...
1
vote
1answer
45 views

Question about polynomials over rings and elements in different rings.

So I'm working on some problems from a book and my proof doesn't seem right to me because I'm not sure if theorems in $\mathbb{Z}$ work since I use an element from $\mathbb{Q}$. Anyhow the question is ...
3
votes
1answer
47 views

Resultant property for Integral Domain

I have seen plenty of proofs that $\textrm{Res}(f,g) = 0$ iff $f$ and $g$ have a common factor not constant. But every time it's assumed that $f$ and $g$ are polynomials over some algebraic closed ...
1
vote
1answer
166 views

Order of group of units of ${\bf Z}_{2015}[X]$

I know that the order of the group of units $U({\bf Z}_{2015})$ is 1440 and so the order of the group of units of the polynomial ring ${\bf Z}_{2015}[X]$ must be at least that because we can view each ...
1
vote
1answer
88 views

Linear operator form of a symmetric matrix

I want to determine what it means for a symmetric matrix to be written in terms of linear operators. Perhaps the key? it looks like the form I am looking for might be the self-adjoint operators? ...
1
vote
2answers
120 views

Show that the commutator subgroup of an abelian normal subgroup and whole group has special form.

Let $A$ be an abelian normal subgroup and $x \in G$. Let $G = AC_G(ax)$ for all $a \in A$, then $$ [A, G] = \{ [a,x] : a \in A \}. $$ Any hints? EDIT: What I have done so far. Let $y = [a, g] \in ...
0
votes
1answer
89 views

The alternating group is a normal subgroup of the symmetric group

For an exercise, I need to prove that the alternating group $A_n$ is a normal subgroup of the symmetric group $S_n$. As clue they say we can use a group homomorphism $\operatorname{sgn} : S_n \to ...
-3
votes
1answer
160 views

By applying the Fundamental Theorem of Homomorphisms, show that there is a ring isomorphism $g: \mathbb{Z}_{4} \to \operatorname{im}(f)$

Please refer to the question here for additional details. Theorem: If $R$ and $S$ are rings and $\phi: R \to S$ a ring homomorphism defined by $g(n+\ker(\phi)) = f(n)$, then $R/Ker(\phi) \cong ...
0
votes
2answers
159 views

finite dimensional $K$-vector space $V$ with linear endomorphism $T$ is a cyclic module over $K[x]$

Problem: Suppose that $V$ is a finite dimensinal $K$-vector space with a linear endomorphism $T$. Then show that i) Associated $K[x]$-module $V$ is such that $V\cong K[x]/(g)$ for some monic ...
5
votes
1answer
93 views

Splitting field and polynomial of minimal degree

Let's assume that we have a splitting field $F$ over $Q$ that is a finite extension. Let $p(x)$ be the polynomial in $Q[x]$ that has $F$ as a splitting field and is of minimal degree. Is it correct ...
3
votes
0answers
151 views

Find the $\ker(f)$ and $\text{Im}(f)$.

Consider the rings $\mathbb{Z}$, $\mathbb{Z}_{4} = \{\bar{0},\bar{1},\bar{2},\bar{3}\}$ and $\mathbb{Z}_{12} = \{[0],[1],[2],...[11]\}$. Define $f: \mathbb{Z} \to \mathbb{Z}_{12}$ by $f(x) = 9x$. ...
0
votes
2answers
77 views

What do Ideals tell you? [duplicate]

So I'm revising definitions of algebra for my exam and I'm wondering what an Ideal actually is? I believe the definition is: $I$ is an ideal of $R$ if $xr,rx\in I$ where $r\in R$ and $x\in I$ ...
4
votes
2answers
140 views

Why an R-module is an R/I-module precisely when it is annihilated by I?

Let $I$ be an ideal of $R$ and let $M$ be an $R$-module. Prove that the formula $(r + I) \cdot m = r \cdot m$ for $r + I \in R/I$, and $m \in M$ makes $M$ into an $R/I$-module if and only if $i ...
0
votes
0answers
26 views

Method to construct non-UFD rings?

You know all the usual examples... $$ \mathbb{Z}[\sqrt{-5}] \; : \; 6 = 2\cdot3 =(1+\sqrt{-5})(1-\sqrt{-5})\\\mathbb{Z}[\sqrt{-7}] \; : \; 8 = 2\cdot2\cdot2 = (1+\sqrt{-7})(1-\sqrt{-7}) \\ \cdots$$ ...
1
vote
0answers
26 views

invertible matrix in extension field implies invertible matrix in original field

I need help for an exercise from Herstein's Topics in Algebra, 2ed in Sec.6.7 Canonical Form: Rational Canonical Form. Since concepts like trace and determinant are not introduced yet, I wonder how to ...
1
vote
1answer
64 views

An unexpected(?) way to identify a 2d vector space with its dual

Let $X$ be a finite-dimensional vector space over the field $\mathbb{R}$. Denote the dual of $X$ by $X^*$. Definition: Let us say that a (not necessarily linear) mapping $x \mapsto x^* : X \to ...
0
votes
1answer
34 views

Polynomial Ring Degree functions

$f(x) = 3x + 2$, $g(x) = 2x^2-3$ in $\mathbb{Z_2}$[x]. What is degree of $f(x)g(x)$? is it simply just the highest degree?
6
votes
4answers
317 views

Why is the group $Z_6$ under addition isomorphic to the group $Z_7^*$ under multiplication?

Why is the group $Z_6$ under addition isomorphic to the group $Z_7^*$ under multiplication? So I'm trying to answer this question: Q. Which of the set are isomorphic to each other? ...
0
votes
3answers
91 views

Group Theory Lemma Proof

Suppose that $(G,*)$ and $(H,\circ)$ are groups and $f:G\to H$ is a homomorphism Prove $f(a^n)=f(a)^n$ $ \forall a\in G, n\in \Bbb{Z}$ . I can't think of smart ways to manipulate the properties of ...
0
votes
1answer
52 views

Quotient Rings of Algebras

So let us take the commutative Banach algebra $B=\mathscr{l}^1(\mathbb{Z}_n)$ over $\mathbb{R}$ with convolution as multiplication $(f\ast g)(x)=\underset{y\in \mathbb{Z}_n}{\sum} f(y)g(x-y)$. I know ...
2
votes
0answers
27 views

Given two arbitrary terms of a geometric series, can this lead to geometric series parameters that cannot be solved for in terms of radicals?

So some higher degree polynomials (and polynomial systems) can still have their roots be solved for in terms of radicals. It comes down to Galois groups and I'm a bit rusty on that subject so I don't ...
2
votes
3answers
54 views

Frobenius injective for finite fields - what about $\mathbb{F_{p^n}}$

Quick question about the Frobenius endomorphism. My lecture notes and wikipedia say that the Frobenius is injective for finite fields. However, if we look at $\mathbb{F_4}$, we have $$\text{Frob}(2) = ...
1
vote
3answers
79 views

Binary Operations, Associative Operations

I'm stuck on this question, please help. The binary operation $*$ is defined on $z$ by $x*y=xy-x-y+c$ for all $x, y, c$ belonging to $\Bbb Z$, $c$ is a constant. Given that $*$ is associative, what ...
5
votes
4answers
96 views

What does $\Bbb{C}(X)$ refer to?

I have from a book (b) Let $E = \Bbb{C}(X)$. Then $\operatorname{Aut}(E / \Bbb{C})$ consists of the maps $X \mapsto \dfrac{aX + b}{cX + d}, ad-bc \neq 0\ldots$ Not sure what $\Bbb{C}(X)$ is. ...
1
vote
0answers
76 views

Exact sequences free abelian groups

I have an exact sequence ending with \begin{equation*} \cdots \rightarrow A \xrightarrow{f_1} \mathbb{Z} \xrightarrow{f_2} \mathbb{Z} \oplus \mathbb{Z} \xrightarrow{f_3} \mathbb{Z} \rightarrow (0) ...
4
votes
4answers
144 views

Why is $\sqrt [n] 1$ not an expression “in radicals” of a root of unity?

In Edwards' Galois Theory, in the chapter on Cyclotomic polynomials, the author devotes a lot of effort to proving that prime order primitive roots of unity can be expressed "by radicals", and gives ...
0
votes
1answer
184 views

If $V$ is a vector space to itself, is the idempotent linear transformation, prove there is a basis such that is a diagonal matrix with all $0$ or $1$

Im trying to solve an algebra homework problem, basically i have: $$\phi:V \longrightarrow V$$ such that $\phi = \phi^2$, where $V$ is finite dimensional. I already proved that $V = \phi(V) \oplus ...
0
votes
1answer
56 views

Faithfully Flat Abelian Groups

I need some help to find faithfully flat abelian groups. Flat abelian groups are torsion free $\mathbb{Z}$-modules. But what about faithfully flat abelian groups. $\mathbb{Q}$ is an example that is ...
1
vote
1answer
59 views

Basis of Q$(\sqrt[4]{3})$/Q

I want to show that Q$(\sqrt[4]{3})$/Q is algebraic. I am pretty sure that {1, $(\sqrt[4]{3})$} is a basis of this extension, and I was wondering if in general, for a (some irrational number) and the ...
1
vote
1answer
170 views

Where do the enormous simple groups come from?

I mean, these simple groups of big order such as 808017424794512875886459904961710757005754368000000000 I think it's order is something similar to a factorial for all those 0s... but I'd like to know ...
1
vote
2answers
59 views

Spaces that satisfy closure with respect to addition and scalar multiplication but aren't vector spaces?

This question is probably a silly one, but I'm not the greatest at coming up with counterexamples. Perhaps someone can lend some insight. The usual (I think?) definition of a vector space $V$ is a ...
2
votes
1answer
48 views

Prove $I \subseteq I+J$ and $J \subseteq I+J$

Let $R$ be a ring and $I$ and $J$ be the ideals of $R$. Prove Prove $I \subseteq I+J$ and $J \subseteq I+J$ I know this is very trivial, but I still need to check what I am doing is correct or not... ...
0
votes
1answer
37 views

Finitely generated as an Algebra

Let $R,S$ be rings. Is the following equivalent to saying $S$ is finitely generated as an $R$-algebra? "For some $n \in \mathbb{N} $ there exists a surjective ring homomorphism from ...
2
votes
1answer
218 views

Galois group for $x^8 - 2$

My textbook asked me to find the Galois group $G$ for $x^8 - 2$. Ok, so the roots of $x^8 - 2$ are $e^{2\pi ik/8} * 2^{1/8}:0 \le k < 8$, by my calculations, so the splitting field is ...
0
votes
1answer
108 views

Why doesn't there exist a ring homomorphism between these two rings?

Why doesn't there exist a ring homomorphism between $\mathbf{Q}[x]/(x^2-2) $ and $\mathbf{Q}[x]/(x^2 -3) $? I see both rings are in fact fields as the polynomials are irreducible, further I know for ...
1
vote
1answer
104 views

Definition of tensor product

Here is the standard dedonition of the tensor product of two modules: Definition for tensor products of two modules: Let $R$ be a ring and $M$ be a right module and $N$ be a left module. ...
0
votes
3answers
73 views

Prove/disprove: $I \cup J$ is (always) an Ideal of $R$.

Let $I$ and $J$ be the ideals of $R$. Prove/disprove: $I \cup J$ is (always) an Ideal of $R$. Rough Sketch: Since, $I$ and $J$ are the ideals of $R$, we have $0_R \in I$ or $0_R \in J$. Hence, $0_R ...
1
vote
0answers
34 views

Simplified Exponent in $\bmod$ equation

I am trying to simplified the following expression: $$\left(y^m\right)^{x} \bmod p$$ In my case, I can only solve $(y^x) \bmod p$ first without prior knowledge of $m$. Eventually, my answer should ...
0
votes
1answer
30 views

Manipulations of Euclidean domains

I am trying to answer the following question For (a) I have said that a and ab are in the ring R, by the definition of a ring. Therefore, by the definition of a Euclidean domain a=abq+r. As we are ...
0
votes
1answer
109 views

Why is the $ \mathbb{Z} $ - Hodge conjecture false?

I wish someone well initiated in the area , told me , why the $ \mathbb{Z} $ - Hodge conjecture is wrong, which allowed to change its state by tensoring by $ \mathbb{ Q } $ to become as it is known ...
0
votes
1answer
25 views

Ring Theory and Modules in Norm

Let $R = Z[\sqrt{5}]$ and $f : R → Z$ be defined as $f(x + y\sqrt{5}) := | x^2 − 5y^2|$. Then prove that $f$ is multiplicative : $f(αβ) = f(α)f(β)$, $∀α, β ∈ R$.
3
votes
1answer
81 views

In a PID , show that a maximal ideal is a prime ideal and conversely.

Suppose that $R$ is a PID and that $I$ is an an ideal. Then $I$ is maximal iff for any $x$ generating $I$, $x$ is irreducible. This is my try Proof: ⇒: Suppose I is maximal and that I is ...