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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Semi projective modules

Consider the quotient field K of a discrete valuation ring R which is not complete. Is R-module M = K^2 is quasi-principally projective (Semi projective).Also M is direct-supplemented and amply ...
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Proof for quotient polynomial rings equivalent to field extension

I am predominantely looking for a proof, I have seen in my books and around but seem to have a hard time finding that if we let $\alpha_1,\alpha_2,...,\alpha_n$ be the roots of the minimal polynomial ...
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about maximal ideals

I don`t understand “if $x∉J$ then $J ⊂ x+J$. Please explain me and show me that every element of $J$ is in $x+J$. $J$ be a maximal ideal and suppose $xy$ is in $J$. We want to show either $x$ is in ...
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1answer
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Quadratic Extensions

I am having a hard time understanding the concept of quadratic extensions. My book explains it: If the minimum polynomial of $a$ over a field $F$ has degree 2, we call $F(a)$ a quadratic ...
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29 views

Irreducible projective cubic, exists continuous surjection?

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...
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If $a_1,a_2,a_3$ are roots $x^3+7x^2-8x+3,$ find the polynomial with roots $a_1^2,a_2^2,a_3^2$

If $a_1,a_2,a_3$ are the roots of the cubic $x^3+7x^2-8x +3,$ find the cubic polynomial whose roots are: $a_1^2,a_2^2,a_3^2$ and the polynomial whose roots are $\frac{1}{a_1}, \frac{1}{a_2}, ...
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What is an embedding of extensions?

I'm given a definition that I don't understand. I just want to have an understanding of it. It goes as follows. We have two Field extensions $H$ and $K$ of a field $F$ and a map $v: K \to H$. They ...
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28 views

Field Extension Question for Polynomials

I cannot seem to find the answer to this question on the internet. It is a question about field extensions for an element $a,b \neq F$ but in some extension $K$. I am wondering if $F(a,b)= \lbrace ...
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Abstract algebra problem on automorphism

I have been going through Herstein's Algebra and came across this problem: "G has an order p^n*m where p is a prime, p doesn't divide m. Suppose G has a normal subgroup P of order p^n. Prove that ...
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Generated subring and finiteness

I need some help with this question: Let $A$ be the subring of $\mathbb{Q}(i)$ generated by $\mathbb{Z}[i]$, $\frac{1}{1+2i}$ and $\frac{1}{(2+3i)}$. Given $n\in\mathbb{Z}$, we can ensure that $A/nA$ ...
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Brilliant formulaes [on hold]

Hey Brilliant mathematician, i am very honored for having your time. I need general Formulas on breaking down a number to a different and being able to derive that number back, my requirements is to ...
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120 views

How to find the irreducible polynomial?

It is giving me a lot of trouble, and I'm beginning to think it's not possible. Find $\operatorname{irr}(2\sqrt{2} + \sqrt{7})$. I start like this: $x = 2\sqrt{2} + \sqrt{7}$ I have squared ...
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1answer
50 views

Necessarily a homeomorphism?

Let $D$ be the projective curve defined by $y^2z = x^3.$ Consider the map $f: \mathbb{P}_1 \to D$ defined by$$f[s, t] = [s^2t, s^3, t^3].$$Is it necessarily a homeomorphism? Any help would be greatly ...
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55 views

Irreducibility of $X(X-3)(X-\alpha)(X-\beta) + 1$

I'm trying to solve the following exercise: Show that for $\alpha,\beta\geq 3$, the polynomial $X(X-3)(X-\alpha)(X-\beta) + 1\in\mathbb Z[X]$ is irreducible. It is straightforward to check that ...
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51 views

Fields of characteristic $p$ that are isomorphic as vector spaces, but not as fields

Give an example of a field $\mathbb{K}$ of characteristic $p > 0$ and elements $a,b \in \bar{\mathbb{K}}$, such that $\mathbb{K}(a)$ and $\mathbb{K}(b)$ are isomorphic as vector spaces over ...
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Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$. I'm particurly struggling with what would be the approach to finding the splitting field of this ...
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Proving existence of unique maximal subfields of Galois extensions with particular properties

A question I am working on asks the following: Let $K / \mathbb{Q}$ be a Galois extension. Prove that there exists a unique maximal subfield $F$ of $K$ such that $F / \mathbb{Q}$ is Galois with ...
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33 views

The degree-genus formula cannot be applied to singular curves in $\mathbb{P}_2$?

(The degree-genus formula) The Euler number $\chi$ and genus $g$ of a nonsingular projective curve of degree $d$ in $\mathbb{P}_2$ are given by$$\chi = d(3-d)$$and$$g = {1\over2}(d-1)(d-2).$$ My ...
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188 views

Can Zorn's Lemma be 'inverted' like this:?

Let $R$ be a (commutative) ring not equal to $0$. I want to show that the set of prime ideals of $R$ has a minimal element w.r.t. inclusion. This may be a wholeheartedly wrong attempt, but I thought ...
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How to solve a congruence using Fermat's Theorem?

I'm reading Fraleigh's A First Course in Abstract Algebra and I'm trying to understand an example (later I have to solve several problems of the same type). Little Theorem of Fermat: If $a\in ...
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42 views

A morphism which is not a comorphism of a regular map

In the lecture, we dealt with morphisms, comorphisms and regular maps. The professor then brought the following example: Let $U$ and $V$ be quasi-affine sets over $\mathbb{C}$ and let $\psi \colon ...
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The number of $3\times 3$ nilpotent matrices over $\mathbb{F}_q$ using the Orbit-Stabilizer theorem

The Fine-Herstein theorem says that the number of of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ is $q^{n^2-n}$. I am trying to verify this for the cases $n=3$ using the orbit-stabilizer ...
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extending isomorphism to an automorphism

Lets say $K/F$ is a field extension and $\alpha ,\alpha '\in K$ are two distinct roots of the same irreducible polynomial in $F[x]$. there exists an isomorphism $$\psi:F(\alpha)\rightarrow ...
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21 views

References for loop rings

I recently saw a paper on alternative loop rings, as always, I am interested in all kinds of rings, and new kind of ring looked attractive. I would like to read loop and then loop rings in detail from ...
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28 views

$\operatorname{Ext}$ and projective dim

I have some problem to understand the proof of proposition 8.38 page 473 of homology by Rotman. Proposition: Let $x\in Z(R)$ be an element which is not a zero-divisor and let $R^*=R/(x)$. Moreover, ...
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Galois group of $ \mathbb{Q}(\varepsilon_5) / \mathbb{Q} $

I'm trying to solve the following problem: Let $ L =\mathbb{Q}(\varepsilon_5) $ be an extension of $ \mathbb{Q} $. Find the Galois group $ G(L / \mathbb{Q})$ $ \varepsilon_5 $ is the primitive root ...
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2answers
92 views

Question about homomorphisms

I have a question that asks the following: Let $S,*$ and $T,.$ be binary structures and let the there be a homomorphism betweeen the two. If this is surjective, then if $S$ is a group, so is $T$. ...
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22 views

Minimal polynomial over $\mathbb{Q}$ of $\alpha$ in $\mathbb{C}$ has coefficients in $\mathbb{Z}$?

Let $m \in \mathbb{C}$ be integral over $\mathbb{Z}$. Prove that the minimal polynomial over $\mathbb{Q}$ has coefficients in $\mathbb{Z}$. The definition I use: $m\in \mathbb{C}$ is integral over ...
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How is $\lbrace a_1, a_2, …, a_n : a_i \in \Bbb Z_2\rbrace$ a group?

I was asked to prove that if we define \begin{equation*} \Bbb Z_2^n = \lbrace a_1, a_2, ..., a_n : a_i \in \Bbb Z_2\rbrace \end{equation*} then it's a group under the operation of addition like ...
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29 views

Proof of second homomorphism theorem

Let $G$ be a group, $H \leq G$ and $N \unlhd G$. Let $HN=\{hn │h \in H, n \in N\}$. I finished (a) $H \cap N \unlhd H$, (b) $HN \leq G$, (c) $N \subset HN$ and $N \unlhd HN$. I have a question in ...
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28 views

What is the number of elements in $Aut(Q(\pi)/Q)$?

I tried to prove that $|Aut(E/F)|$ is finite, then $E/F$ is a finite extension, but then now I think $Q(\pi)/Q$ would be a counterexample for this. I can see that there are two automorphisms ...
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26 views

Show that $N(H):=\{g\in G; gHg^{-1}=H\}$ is subgroup of $G$

Let $G$ be a group and $H$ subgroup of $G$, $N(H):=\{g\in G; gHg^{-1}=H\}$ I need to prove that $N(H)$ is subgroup of $G$. It's almost the same question like :$\forall g \in G, gHg^{-1} = H ...
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33 views

If $R[X]$ is ED then $R[X] $ is PID

Is this true and why. If $R[X]$ is ED then $R[X] $ is PID . Thanks for help.
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1answer
55 views

Exact functors preserve free modules?

Let $R$ be a principal ideal domain. Suppose that $F$ be an exact functor from the category of $R$-modules to itself. If $M$ is a free $R$-module, is $F(M)$ still a free $R$-module? I do not know how ...
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Order of prime ideals over split primes in the class group.

Let $P$ be a prime ideal of $\mathcal O_K$ ($K$ a quadratic field) and let $P$ have norm $p$ where $p$ is a split prime. Is it possible for the ideal class $[P]$ to have order less than three? I feel ...
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64 views

Multiplicative Inverse Element in $\mathbb{Q}[\sqrt[3]{2}]$

So elements of this ring look like $$a+b\sqrt[3]{2}+c\sqrt[3]{4}$$ If I want to find the multiplicative inverse element for the above general element, then I'm trying to find $x,y,z\in\mathbb{Q}$ such ...
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21 views

Prove this result relating to the sign of a permutation

Suppose that $\phi \in S_n$ is a permutation. Suppose also that $\psi = \phi \circ (i,j),$ where $1 \leq i, j \leq n.$ Why does it follow that sign$(\phi) = $ $-$sign$(\psi)$?
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Show that If $R[X]$ is Euclidean domain then $R$ is a field [duplicate]

Let $R$ is an integral domain . Show that If $R[X]$ is Euclidean domain then $R$ is a field . I'll be waiting for your help. Thank you very much in advance!
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1answer
24 views

Noetherian ring of symmetric polynomials

I wish to show that $k[x_1,x_2,..,x_n]^{\Sigma_n}$, which is the ring of all symmetric polynomials, is Noetherian. I thought the easiest way to do this would be to show that every ideal is ...
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If $K$ is a complex of $R-$modules and $J$ is an injective $R-$module, prove that $H^n(Hom_R(K,J))\cong Hom_R(H_n(K),J)$

If $K$ is a complex of $R-$modules and $J$ is an injective $R-$module, prove that \begin{equation*} H^n(Hom_R(K,J))\cong Hom_R(H_n(K),J). \end{equation*} Thank you in advance.
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Is the congruence relations lattice of a lattice a sublattice of all equivalence relations on it?

By this Wikipedia link, it seem the set of all congruence relaions on a lattice $(X,\le)$ is a complete lattice with inclusion. Is this lattice a (complete) sublattice of the lattice of all ...
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88 views

Verifying if a multiplication table is from a group

I'm asked to verify which of these multiplication tables form a group. I'm having problems to see which of the axioms for a group are violated in each table. In (a), I couldn't find an element $e$ ...
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How to find the Galois group of a given polynomial from Galois viewpoint?

Excerpt from Page 3 Consider the polynomial equation \begin{equation*} x^4 - 2 = 0, \end{equation*} with integer coefficients. We have four solutions: \begin{equation*} \alpha = ...
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Is this a valid way of solving modular equations?

Some of the exercises in my abstract algebra textbook involve solving congruence equations, for example $$5x \equiv 1 \pmod 6$$ I convert that to a linear equation by rewriting $$x = \frac{k6+1}{5} = ...
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Reference Request: Replacement for Anderson and Fuller

I'm working through Rings and Categories of Modules by Anderson and Fuller. It's a great text, but I would like a more modern replacement. The text focuses too much on the language of generation and ...
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Problems related to field theory [on hold]

Suppose that $ F $ is a field whose characteristic is not 2. If nonzero elements of $ F $ form a cyclic group under multiplication then show that $F $ is finite
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Structure of the group $\{1+p\mathbb Z_p \}$

In preparation for algebraic number theory I am reading Serre : A course in Arithmetic. I stuck in understanding a proof (p.17): Notation: $U_n=1+p^n\mathbb Z_p$ Actually there are many things ...
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1answer
50 views

Show every $a \in E^*$ is a root of $x^{p^d-1} -1 $?

Let $\mathbb{Z}_p < E$ be an extension field of degree $d$. A simple counting argument shows: $|E^*| = p^d - 1$ Proposition: For all $\alpha \in E^*$, $x^{p^d-1} -1 = 0.$ In a field of $p^d ...
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28 views

Quadratic residue and permutation [on hold]

Let $p>2$ be a prime number and $a \in \mathbb{Z}_{p}$. For an integer $k$ consider the permutation $\pi$ of the set defined by $\pi: n \to kn+a \pmod p$. Prove that $k$ is quadratic residue modulo ...
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Etingof problem 2.15.1 Representations of sl(2)

I'm studying from Etingof's Introduction to Representation Theory. This is problem 2.15.1, part a. I feel I'm close to the solution. Here's what I have. Problem: A representation of sl(2) is a vector ...