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 do I prove that $X^{p^n}-X$ is the product of all monic irreducible polynomials of degree dividing $n$?

How do I prove that $X^{p^n}-X$ is the product of all monic irreducible polynomials in $\mathbb Z_p[X]$ of degree dividing $n$? Let $\bar Z_p$ be an algebraic closure of $Z_p$. Define $F=\{x\in ...
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1answer
34 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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2answers
56 views

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

If $R$ is a commutative 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). If $n\geq 1$, is there an example ...
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51 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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1answer
35 views

For a $p$-group $G$, and $H \le G$, if $G=H G^\prime$, then $H=G$ [closed]

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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1answer
144 views

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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2answers
35 views

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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1answer
52 views

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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25 views

Conjugacy classes of solvable groups [closed]

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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26 views

Prove that if G is finite, then |H ∩ K| is a divisor of |H| and a divisor of |K|. [closed]

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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0answers
16 views

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 $\implies$ ...
2
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2answers
49 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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1answer
59 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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1answer
50 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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1answer
28 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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2answers
46 views

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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1answer
15 views

$A$ 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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1answer
16 views

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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23 views

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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2answers
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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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1answer
64 views

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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0answers
31 views

Degree $5$ Irreducible Polynomial Solvable by Radicals and Abelian Extension

Consider an irreducible polynomial of degree $5$ over $\mathbb{Q}$ which is solvable by radicals. How do I show that its splitting field is contained in a field of the form $K(a)$, where $K$ is an ...
2
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1answer
35 views

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 ...
2
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1answer
53 views

$\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$, but $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ is not

How can I show that the extension $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ but that $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ is not? I am kind of lost with Galois Theory. Thanks
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2answers
21 views

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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4answers
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Can you check my proof of Fermat's Last Theorem? [closed]

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 ...
3
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1answer
81 views

$x^p-x-1$ 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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0answers
41 views

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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3answers
58 views

$\mathbb Z[1/2]$ 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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1answer
44 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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1answer
29 views

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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2answers
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Why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$?

I do not understand why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$. I know that $x^{4}+1$ is irreducible $(f(x + 1) = (x + 1)^4 + 1$ is Eisenstein at $2$) and it has the roots: ...
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1answer
28 views

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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0answers
29 views

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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1answer
30 views

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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1answer
30 views

Galois group of $f$ is cyclic if $\deg f$ is prime

Hello I am learning Galois Theory by myself and got lost in the following exercise: Let $f$ be an irreducible polynomial of degree $n$, and suppose that the splitting field of $f$ is generated by a ...
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1answer
18 views

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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1answer
47 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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1answer
41 views

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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29 views

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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1answer
37 views

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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1answer
32 views

Provide an example to show that $S$ may not necessarily be a unique factorisation domain when $R$ is a unique factorisation domain.

Let $R$ and $S$ be integral domains, and suppose that $\phi:R \rightarrow S$ is a surjective ring homomorphism. Provide an example to show that $S$ may not necessarily be a unique factorisation domain ...
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2answers
54 views

Show that $Im(\phi) = \mathbb{Z}[i]$

Let $\phi: \mathbb{Z}[x]\to \mathbb{C}$ and $\phi(f(x)) = f(i), \forall f(x) \in \mathbb{Z}[x].$ Show that $Im(\phi) = \mathbb{Z}[i]$ My attempt: I am not sure if it's correct: First, we need to ...
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0answers
15 views

Show that the positive units of $K$ form a group isomorphic to $\mathbb Z$

Let $K$ be a cubic field such that $r_1=r_2=1$. Suppose $K$ is imbedded in $ \mathbb R$ ($r_1$ and $r_2$ are the usual notations, $r_1$ denotes the number of isomorphisms $\sigma : K \to \mathbb R$ ...
3
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2answers
30 views

How do I prove that $\forall \beta\in F(\alpha)\setminus F$ is transcendental?

Let $E/F$ be a field extension. Let $\alpha\in E$ be transcendental over $F$. Let $\beta\in F(\alpha)\setminus F$. Then, how do I prove that $\beta$ is transcendental over $F$? Here's how I tried: ...
2
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3answers
69 views

How many elements does the free product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ have?

Taken from Hatcher, I think the free product $G =\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ should have infinite elements taking the form of words containing alternate elements, i.e. $a, b, ab, ...
3
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0answers
56 views

“Breaking the symmetry” in solving algebraic equations

I've heard somewhere a discussion about solving algebraic equations before: When solving a quadratic equation, we are essentially doing the following. Observe that ...
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0answers
42 views

How many elements are in $\mathbb{Z}_5[i]/\langle1+i\rangle$

Maybe this question has been asked by someone else before, but I could not find a duplicate and I would really appreciate some help. I know that an element of $\mathbb{Z}_5[i]/\langle1+i\rangle$ is ...