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 classify the rings of fractions of a principal ideal domain?

Let $A$ be a principal ideal domain and let $K$ be its field of fractions. I proved a) Every ring $B$ such that $A \subset B \subset K$ is a ring of fractions of $A$, and c) Show that any ring of ...
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$x^{p^{k}-1}-1$ divides $x^{p^{n}-1}-1$ in $\mathbb{F}_{p}$ iff $k$ divides $n$

Let's consider two polynomials in $\mathbb{F}_{p}[x]$: $f(x)=x^{p^{n}-1}-1$ and $g(x)=x^{p^{k}-1}-1$. How to prove that $g(x) \mid f(x)$ iff $k \mid n$? For instance, it's worth trying this: ...
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How to prove this is a field?

Let $F=(\mathbb{Z}/5\mathbb{Z})[x]/(x^2+2x+3)$. How do I prove $F$ is a field? I've shown its a commutative ring with an identity $\bar1$. Then we let ...
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Determining all the subfields of the splitting field for $x^8-2$ which are Galois over $\mathbb{Q}$

I have found all the subgroups of the Galois group of the splitting field $K=\mathbb{Q}(\sqrt[8]{2},i)$ over $\mathbb{Q}$, and I know that if any of the subgroups $H$ of $G=$Gal$(K/\mathbb{Q})$ are ...
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Let $R$ be a PID and $I$ a prime ideal of $R$ s.t. $0 \subset I \subset 1_R$

Let $R$ be a PID and $I$ a prime ideal of $R$ s.t. $0 \subset I \subset 1_R$ and let $I = \langle a \rangle$, where $a$ is a prime element of $R$. My question is: is there any other prime ideal $J$ ...
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Let G be a group. Use the FHT (Fundamental Homomorphism Theorem) to prove that the quotient group G/{e} is isomorphic to G.

Ok so the Fundamental Homomorphism Theorem (or First Group Isomorphism Theorem) states that if θ : G → H and ker (θ) = K, then the quotient group G/K is isomorphic to H. I know that θ : G → G has to ...
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Prove that $\Bbb{Z}[i]/I$ is finite where I is an ideal of $\Bbb{Z}[i]$

Show that for any nontrivial ideal $I$ of $\Bbb{Z}[i]$, $\Bbb{Z}[i]/I$ is finite. $\Bbb{Z}[i]$ is a PID, so $I=\langle{a+ib\rangle}$. Now $\Bbb{Z}[i]/I$ has elements of the form ...
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Can a non-commutative ring contains identity?

Can a non-commutative ring $R$ contains identity? Suppose $R$ contains the identity element 1. Construct an ideal $Z(R) = \{a \in R \mid ra = ar\text{ for all }r \in R\}$. Since $1 \in Z(R)$, $R = ...
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Proof of primitive element theorem for $F$ finite

I am trying to understand the proof of primitive element theorem, in particular this statement: Let $E/F$ be a field extension such that there are finitely intermediate fields containing $F$. Then $E ...
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Can icosahedral group have subgroup of order 30?

The textbook (Artin) wants me to show that there is no subgroup of order 30 in the icosahedral group. I tried to use the index(2) of the subgroup of order 30, but I can't get any new ideas. What other ...
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Modules over Itself

Let $F$ be a field and let $R$ be the ring of polynomials over $F$ in infinitely many variables $x_1, x_2 , \cdots x_n \cdots$ (Of course, each element of $F$, being a polynomial, will involve only ...
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25 views

How do I know an element generates a coordinate ring K[W] as a vector space over K?

I have an example which proves that a cuspidal cubic $W\subset \mathbb{A}^2$ defined by $y^2-x^3=0$ is not isomorphic to to $\mathbb{A}^1$. I'll start by defining a few things: Let $V=\mathbb{A}^1$, ...
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28 views

Direct sum of two non-zero $R$-modules

If $R$ is a commuative integral domain, how can I show that it cannot decompose as a direct sum of two non-zero $R$-modules (when viewed as a module over itself). Now if $n\geq 1$, is there an example ...
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47 views

Prove $\mathbb{Z} [\sqrt{2}]$ is a Euclidean ring.

The definition of Euclidean ring: An integral domain R is called Euclidean ring if $\exists \delta$ : $R${$0$} -> $\mathbb{N} \cup{0}$ satisfying: (1) $\delta (a) \leqslant \delta (ab)$ if a, b $\in ...
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For a $p$-group $G$, and $H \le G$, if $G=H G^\prime$, then $H=G$ [on hold]

For a $p$-group $G$, and $H \le G$, prove that, if $G=H G^\prime$, then $H=G$; where $G^\prime$ is the commutator subgroup of $G$.
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Is this element algebraic over $\mathbb{Q}$?

I'm trying to see whether the following element is algebraic over $\mathbb{Q}$ and if it is - try and find its degree: $$ a = \left(\frac{(1 + \sqrt[3]{7})^{7/5}}{(\sqrt[3]{7} - 7)^3 + ...
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Subgroup $H \leq G$ acting on $G$ by translation is transitive?

In Elementary Topology. Textbook in Problems, by Viro, et al they state the following: Let $G$ be a topological group, $H \leq G$ a subgroup. Then $G$ is a homogeneous $H$-space under the ...
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Example such that $ f_1(x)$ is reducible but $f(x)$ is irreducible.

(The (mod p) Irreducibility Test) Let $p$ be a prime an suppose that $f(x) \in \mathbb Z[x]$ with $\deg f(x) \geq 1$. Let $f_1(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by ...
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Show that the free product of countably many countable groups is countable.

The question I am struggling with is as follows: Suppose that $\{G_\alpha\}$ is a countable collection of countable groups. Show that $\ast_{\alpha}G_\alpha$ is countable. The definition of ...
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Conjugacy classes of solvable groups [on hold]

If $A$ be a subset of solvable group $G$, let $k_G(A)$ be the number of $G$- conjugacy classes contained in $A$. Also let $N$ be a normal subgroup of $G$ of odd order, $\frac{G}{N}$ be an abelian ...
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Prove that if G is finite, then |H ∩ K| is a divisor of |H| and a divisor of |K|. [on hold]

H and K are subgroups of a group G, and H ∩ K is a subgroup of H and a subgroup of K ===Prove that if G is finite, then |H ∩ K| is a divisor of |H| and a divisor of |K|. =====If |H| = 28 and |K| = ...
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Help in the proof of $k \subset F$ is radical if and only if it is solvable.

Let $k \subset F$ be a Galois Extension, with char $k=0$. Provided it has enough roots of $1$, $k \subset F$ is radical if and only if it is solvable. I am trying to show that Solvability $\implies$ ...
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35 views

Primitive element and Galois group Computation

Let $L=\mathbb{Q}(\sqrt{2},\sqrt{3},y)$ where $y^2=(9-5\sqrt{3})(2-\sqrt{2})$. Show that $y$ is a primitive element of the extension $L/\mathbb{Q}$. Determine its minimal polynomial and a splitting ...
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47 views

Writing equations without latex (for labelling, classification, structure analysis and rendering according to mathematical meaning)

I'm searching a solution to write equations other than (but compatible with) latex. While latex is excellent for math rendering, it is not suited to describe the structure or meaning of the equations. ...
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45 views

Finite conjugate subgroup

In a paper titled "Trivial units in Group Rings" by Farkas, what does it mean by Finite conjugate subgroup. Here is the related image attached- What is finite conjugate subgroup of a group? It is ...
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19 views

Construct the Normal Closure for this extension

I am trying to find the normal closure for the following extension $\Bbb Q\subset\Bbb Q(t)$, where "$t$" is a zero of $x^3-3x^2+3$ and $\Bbb Q$ are the rational numbers. I know the normal closure ...
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If K and H are normal subgroups of $G$, $H \cap K = \{1\}$ and both $G/H$ and $G/K$ are abelian, then $G$ is abelian.

Let G be a group, and $H \trianglelefteq G$, $K \trianglelefteq G$. Prove that if $H \cap K = \{1\}$ and $ G / H $ and $ G/ K $ are abelian, then G is abelian. I've tried to give a proof by ...
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$A$ prime in $S$ implies that $\phi^{-1}(A)$ prime in $R$ ; $A$ maximal in $S$ implies that $\phi^{-1}(A)$ maximal in $R$

Suppose $R,S$ are commutative rings with unities. Let $\phi$ be a ring homomorphism mapping $R\to S$ and let $A\subset S$ be an ideal. How can I start the proofs for: Showing that $A$ prime in ...
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Logical comparison of two values with algebra

Suppose I have two real numbers A and B (A $\wedge$ B $\subset$ $\mathbb{R}$). I want to do some algebra over these number and get 1 if they are equal and get 0 if not. For example: In this ...
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Ordering rational functions in a field

This is from Fraleigh, A First Course in Abstract Algebra, 7th edition, Section 25, "Ordered Rings and Fields", questions 10, 11,12, and 13. Here's the question: List the given elements in an order ...
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If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$.

Herstein 3.4.20: If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$. I don't understand why $\varphi$ needs ...
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Question about of the polynomial $x^p -x -a$

If $F$ a field with $char(F)=p$. Prove: If $x^p -x -a$ is reducible in $F[x]$ , then this it splits in distinct factors in $F[x]$. I know if for hypothesis $x^p -x -a = P(x)Q(x)$ with $P(x),Q(x) ...
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Proving that the unit cube cannot be tripled (with straight edge and compass)

I would like to show the unit cube cannot be tripled using a straightedge and compass. I note that the side of a cube that has been tripled would have a side length of $\alpha=\sqrt[3]{3}$ But ...
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The number of $q$-Sylow subgroups cannot be $p$ for prime $p<q$

Since $q>p $, we cannot have $n_q=p $. Here $n_q $ is the number of $q $ Sylow subgroups. Why is the above statement true? This is a statement from Dummit and Foote.
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Can you check my proof of Fermat's Last Theorem? [on hold]

I've come up with a proof of Fermat's Last Theorem and my teacher would not look at it so i was wondering if you could check. I know it's supposed to be hard to prove, but I use a "trick" from calc ...
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p-primary component of a group

I have been asked to find the $3$-primary component of the group: $$\mathbb{Z_3}\oplus\mathbb{Z_5}\oplus\mathbb{Z_9}\oplus\mathbb{Z_{153}}$$ Now, I know that we define the $p$-primary component of a ...
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$x^p-x-1$ is irreducible over $\mathbb{Q}$[x]

For any prime p, prove that $x^p-x-1$ is irreducible over $\mathbb{Q}$[x]. (In a field of characteristic p this is true). I asummed exist root in $\mathbb{Q}$, let's call $\frac{\alpha}{\beta} ...
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Proving the Third Isomorphism Theorem

If $M$ and $N$ are normal subgroups of $G$ and $N$ $\leq$ M, prove that ($G$/$N$)/($M$/$N$) $\approx$ $G/M$. What I have so far: We can define a mapping $\phi$: $G/N$ $\mapsto$ $G/H$ by ...
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$\mathbb Z[1/2]$ is not finitely generated?

$\mathbb Z[1/2]$ is not finitely generated ? Maybe I misunderstood, what finitely generated means, here http://en.wikipedia.org/wiki/Finitely-generated_module#Formal_definition, it says, we need ...
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37 views

Ring Homomorphisms from a ring R to a ring S.

The book I am using for my Abstract Algebra course is Contemporary Abstract Algebra by Joseph A. Gallian. Find the rings $R$, $S$ as below. There is a ring homomorphism from a ring $R$ to a ring ...
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Group of isometries in a plane

Which elements of the group of all isometries of $\mathbb{R} ^2$ are squares? (i.e. $g=h^2$ for $h \in G$).
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Determine the Kernel and Homomorphisms

Suppose that $\phi$ is a homomorphism from $S_{4}$ onto $Z_{2}$. Determine Ker $\phi$. Determine all homomorphisms from $S_{4}$ to $Z_{2}$. What I have so far: We know by the First Isomorphism ...
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Cycle index for tetrahedron

Calculate the cycle index for the group of all symmetries of a regular tetrahedron. Is my solution correct? Since group of all symmetries of a regular tetrahedron is the group of permutation $S_4$ ...
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What is the discriminant of $R:=\mathbb Z[\sqrt{2},\sqrt{3}]$?

What is the discriminant of $R:=\mathbb Z[\sqrt{2},\sqrt{3}]$ ? The discriminant is defined as the determinant of the matrix $\left(tr(x_ix_j)\right)_{1\le i,j\le n}$ for any basis ...
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Minimum possible size of generating set of $(\mathbb{Z}_p)^m$

Is it true that the group $(\mathbb{Z}_p)^m$ cannot be generated by less than $m$ elements?
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42 views

Even permutation in Sn

How to show that if $\pi \in S_n$ is a square then $\pi$ is an even permutation. Is the converse statement true: each even permutation is a square?
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Square element in a cyclic group

Which elements of a cyclic group are squares (an element $g$ of a group $G$ is a square if $g=h^2$ for some $h \in G$)? Here is my solution; is it correct? Let $G = \{ 1,a,a^2, \ldots , a^n \}$ ...
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Motivation For Tensor Product of R-Modules

I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important ...
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Squares in a field with $q^n$ elements.

Let $F$ be a field with $q^n$ elements, where $q$ is an odd prime. Write $q^n=2m +1$ with $m \in \mathbb{N}.$ If $r \in F^{\times},$ show that the equation $y^2= r$ has a solution iff $r^m=1.$
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Riddle: Assigning Students into Groups

Suppose you had a classroom with 25 students. You want to assign 6 homework assignments over the course of the term and for each of these assignments students will work in groups of 5. But you want to ...