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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About the group of units

I'm stuck at this section of the following problem: Let $R$ be a commutative ring and $S$ be a subset of $R$, with $S$ multiplicatively closed. Find the group of units of $S^{-1}R$. My try: ...
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Maximal left ideals $\leftrightarrow$ simple left modules

Suppose $R$ is a ring with unity. This passage in Lang's Algebra discusses the correspondence $$\text{Maximal left ideals of $R$} \leftrightarrow \text{Simple left $R$ modules},$$ where I corresponds ...
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Question on intersection of ideals.

Consider the polynomial ring $k[x_0,x_1]$, and the two ideals $I=(x_0,x^2_0 x^2_1,x^3_1)$ and $J=(x^2_0,x^2_1)$. What is the intersection of these ideals? I found that $I \cap J = ...
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1answer
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Why can't Eisenstein Criterion be used for certain polynomials (to show that it's irreducible over $\mathbb{Q}$)?

Why can't Eisenstein's Criterion be used to show that $$4x^{10} - 9x^{3} + 21x - 18$$ is irreducible over $\mathbb{Q}$? I mean even if we were to apply Eisenstein here, there doesn't exist a prime ...
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Notation Question(Abstract Algebra)

what does $\mathbb{Q}(\sqrt{3})$ mean?
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Finding primary decompositions of ideals

I have been given this example of the decomposition of an ideal into primary ideals $$ I =⟨x^2,xy,x^2z^2,yz^2⟩$$ Then the primary decomposition of this ideal is: $$⟨x^2,y⟩∩⟨x,z^2⟩⊆K[x,y,z]$$ This ...
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1answer
34 views

Algebra notation $\mathbb{F}^*_p$

Let p be a prime number which is odd. (a) Show that $\mathbb{F}^*_p$ has a unique subgroup of order $(p - 1)/2$, namely the subgroup consisting of squares of elements of $\mathbb{F}^*_p$. My ...
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Visualising rotations of a 3d object

In abstract algebra we are looking at the issue of orbits and stabilisers and applying this to colouring. I am extremely comfortable with the material except for visualising the various rotations of ...
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42 views

Questions about homomorphisms?

I'm running into algebra using homomorphisms for A LOT of things, but I don't think I have a full understanding of what they are. I have read on this site a good explanation that was really great, but ...
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2answers
45 views

What exactly is the purpose of the evaluation homomorphism?

I just don't understand the point of terming the evaluation of a polynomial by a map like this? And what's more, the map is going into a larger field than the field the polynomial is in anyway. What ...
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0answers
29 views

Find all $(x,y,z)\in \Bbb Z^3$ so that $ x^y=x(\text{mod }z)$

I got this problem for Integer Triplets I tried this way : $ x^y=x(\text{mod }z)$ which implies $ x^y-x=0(\text{mod }z)$ and then i got stuck, What should I do? is their a better way
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Determining orbits and stabilisers

It is given that $$(x,y) \cdot \left [\begin{array}{cc} a & b \\ b & a \end{array}\right] =(ax+by,bx+ay)$$ defines an action on the group $$G=\left\{\left [\begin{array}{cc} a & b \\ b ...
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9answers
104 views

Prove/Disprove: if $x^2 = a^2$, then $x = a$

From Prof. Charles Pinter's A Book of Abstract Algebra's Chapter 4 exercises: For each of the following rules, either prove that it is true in every group $G$, ...
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2answers
43 views

Ring homomorphism from $Z/12Z$ to $Z/20Z$?

I've seen the very problem asked previously here but I'm here for more of an explanation for what my textbook is doing. I am asked to find the number of ring homomorphisms from $Z/12Z$ to $Z/20Z$. ...
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1answer
35 views

Solve Group Equation Given Two Equations

Given the group equations $x^{2}=a^{2}$ and $x^{5}=e$, I attempted to solve for $a$ by substituting $a^{2}x^{3}$ into $x^{5}=e$. Then I solved for $x$ by noting $a^{2}x^{3} = e \implies ...
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1answer
13 views

Do matrices have average and fluctuations?

Given a set of numbers, one can calculate the average of those numbers and the fluctuation (variance) over the average. E.g,, $\langle A \rangle=\frac{1}{N}\sum_{i=0}^N A_i$ and $(\delta A)^2 = ...
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2answers
34 views

Show that in a right artinian ring $R$, every prime ideal is a maximal ideal.

Show that in a right artinian ring $R$, every prime ideal is a maximal ideal. **Comments:**I have as a result: For any ring R, are equivalent: (1) R is semisimple; (2) R is semiprime and left ...
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1answer
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Field extensions and algebraic elements

Can somebody explain why taking beta gives $K(\beta)$ as a subspace of $K(\alpha)$?
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4answers
112 views

Are all groups algebraic?

I know the definition of a group as a set with an operation that satisfies certain axioms. I have heard that there is something called an algebraic group and that this is a group with a topology such ...
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2answers
43 views

Field that is a subfield of own of its subfields

Let $K$ and $L$ be fields. We have homomorphisms $f: K \to L$ and $g: L \to K$. Are $K$ and $L$ necessarily isomorphic?
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1answer
47 views

Proving that an element generates $\mathcal{O}_K^*/ (\mathcal{O}_K^*)^3$

Let $K = \mathbb{Q}[x] / \langle x^3 + 2x^2 + 6x + 6\rangle$. The polynomial has a single real root and its discriminant is $-588$. Let $\alpha$ be the image of $x$ in $K$. Then how would I show that ...
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1answer
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Automorphisms (in the context of Galois Theory)

Can someone give an explanation of what an automorphism is, in the context of Galois Theory? I keep thinking it is the set of maps which send roots of polynomials to their conjugates but I feel that ...
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0answers
30 views

The complement of the image of the zero section is still a $\mathbb{G}_m$-torsor?

This came up while doing some reading Schneps text on Galois Groups and Fundamental groups, but it's glossed over. In any case, suppose that you have a line bundle over a scheme $L\to X$, with zero ...
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1answer
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Does all splitting fields have characteristic 0?

Does all splitting field have characteristic 0? This may be a bad question, but I am wondering because the author of a book I am reading summarises a lot of properties when he is about to start with ...
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67 views

Is such set a group?

Question: If a set $G$ is equipped with an associative binary operation $\ast$, and assume $G$ has identity element $e$ and for each $g \in G$ there exists its inverse element $g^{-1}$, is $G$ a ...
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Is the commutator subgroup functor exact?

I'm wondering whether the commutator subgroup functor on the category of groups is an exact functor in the sense that it preserves exact sequences of the form $A \rightrightarrows B \to C$ where the ...
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2answers
38 views

A subgroup of $\textrm{GL}(3,q)$ of order $q^2(q-1)$

Let $q$ be a prime power. Consider the multiplicative group $\textrm{GL}(3,q)$ of the $3 \times 3$ matrixes with coefficients in $\mathbb{F}_q$ which are invertible. The matrixes $$ M_{a,b,c} = \left( ...
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1answer
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Number of ring homomorphisms form $\mathbb Z[x]$ to $\mathbb Z_{12}$

I have tried : Let $f$ be an homomorphism form $\mathbb Z[x]$ to $\mathbb Z_{12}$. we have to find the possible image of $1$ and $x$. Suppose $f(1) = a$, then $f(1)^2 = f(1) = a^2 = a$, then the ...
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3answers
41 views

Simple Solving Equations in Groups Problem

From Prof. Charles Pinter's A Book of Abstract Algebra's Chapter 4 exercises: Let $a, b, c$ and $x$ be elements of a group $G$. In each of the following, solve for $x$ in terms of $a, b$ and $c$. ...
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1answer
45 views

$S_4$ is not supersolvable? Why am I wrong?

I read that $S_4$ is an example of a solvable group who is not a supersolvable group. In order to prove it is solvable, we see that: $\{e\}<\{(1),(12)(34)\}<K<A_4<S_4$ where $K$ is the ...
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Trouble constructing $\mathbb Z_3[x]/(x^2+1)$

If I have $\mathbb Z_3[x]/(x^2+2x+2)$, I can construct a field by letting $x^2=x+1$. The reps are: $0$ $1$ $x$ $x^2=x+1$ $x^3=x^2+x=x+1+x=2x+1$ $x^4=2x^2+x=2x+2+x=2$ $x^5=2x$ $x^6=2x^2=2x+2$ ...
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1answer
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Unique factorization in fields

Suppose $A$ is a commutative $R$-algebra and that is also a field. Define: For $x,y \in A$, say that $x$ divides $y$ iff $xr = y$ for some $r \in R$. Call $x,y \in A$ associates iff each divides the ...
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1answer
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Show the group isomorphism $(\mathbb{Z}/n)^\times \cong (\mathbb{Z}/p_1^{k_1})^\times \times \cdots \times (\mathbb{Z}/p_n^{k_n})^\times$

When $r$ and $s$ are relatively prime we have the ring isomorphism $\mathbb{Z}/rs \cong \mathbb{Z}/r \times \mathbb{Z}/s$ Given a prime factorization of $n$ where $n = p_1^{k_1} \cdots p_n^{k_n}$ ...
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0answers
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Question about principle ideals and polynomials and quotient ring construction?

Say I have a ring of polynomials in $R[x]$. I wish to define the quotient group $R[x]/<x^2+1>$. My question lies in the ideal generated by $<x^2 + 1>$. This is the set of all numbers such ...
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Determine Isomorphism type

Determine isomorphism type of quotient group $$\mathbb{Z} \times \mathbb{Z} / \langle(1,1)\rangle $$ using Fundamental Theorem Finite Generated Abelian Groups after looking at the factor group, it ...
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$p \in C - D$, inflection point for $C$ iff inflection point for $C \cup D$.

Show that if $C$ and $D$ are projective curves in $\mathbb{P}_2$ and $p \in C - D$ then $p$ is a point of inflection for the curve $C$ if and only if $p$ is a point of inflection for the curve $C \cup ...
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Same factors in a normal series

I have the following assertion: Let $G,H$ be finite groups. If they have the same factor groups in a normal series, then they have the same composition factors. I don't know if I understand it well. ...
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6answers
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Prove this polynomial falls within $\mathbb R[x]$

[ The problem below is from Yao Musheng (姚慕生), Wu Quanshui (吴泉水), Advanced Algebra (高等代数学) Ed $2$, Fudan University Press, page $207$. ] Suppose $f(x)\in \mathbb C[x]$. If $\forall c\in \mathbb ...
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Suppose that C is a companion matrix.why F is subfield F[C]?

Suppose that $C$ is a companion matrix of some irreducible monic polynomial $m\in F[x]$ ,$m(x)=m_0+m_1x+…+m_{n−1}x_{n−1}+x_n$. Consider the surjective homomorphism $F[x]→F[C]$ defined by $x↦C$. The ...
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1answer
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Group Theory-Isomorphisms

Currently in Abstract Algebra, discussing group theory. In order to show two groups are isomorphic to each other, I know what you need to show, $1$-$1$, onto, and homomorphism. what I'm having a ...
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1answer
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Question about maximal ideals in a Polynomial ring

I'm reading Freiligh and he has an example in a book, here it is: Let $F = R$ and let $f(x) = x^2 + 1$. Which is well known to have no zeros in $R$ and thus is irreducible over $R$ by a theorem ...
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1answer
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Solving Mordell Equations

I am looking at the solution provided in my lecture notes for solving this particular mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3 $$ In the ...
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1answer
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How to Formulate this Linear Algebra Fact in a Coordinate Free way?

There is a result result given in the last paragraph of pg 15 in Hoffman And Kunze's Linear algebra (2nd Edition) which essentially says that THEOREM. Let $F_1$ be a subfield of a field $F$. If ...
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1answer
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When is composition in $A$-mod surjective?

Let $k$ be any field. Suppose that $A$ is a finite dimensional associative $k$-algebra. Let $A$-mod be the category of finite dimensional (over $k$) left $A$-modules. Then composition of ...
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2answers
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There is an element, whose order is the exponent of $H$

If $H$ is a subgroup of $K^*$, where $K$ is an arbitrary field, then there is an element $h\in H$, whose order is the exponent of $H$, that is the least common multiple of the elements of $H$ I ...
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1answer
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Suppose $\mathcal{C}$ is a category, is it true that if a composition $f\circ g$ of two morphisms is an epimorphism, then $f$ is an epimorphism?

In my "Introduction to Category Theory" class, my teacher wrote on the board something like this: "... due to the fact that if a composition $f\circ g$ of two morphisms is an epimorphism, then ...
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Prove that a functor $F:C_{(X,\leq)}\to C_{(Y,\leq)}$, being $(X,\leq),(Y,\leq)$ partially ordered sets, is just an application that is monotone.

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
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2answers
150 views

Maximum number of Sylow subgroups

I've been studying Sylow-$p$ subgroups, and I've come across this problem. Let $G$ be a finite group. Show that the number of Sylow subgroups of $G$ is at most $\frac{2}{3}|G|$ . ($|G|$ is the ...
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1answer
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Show that the number of reduced residues $a \mod m$ such that $a^{m-1} \equiv 1 \mod m$ is exactly $\prod_{p \mid m} \gcd(p-1,m-1).$

Show that the number of reduced residues $a \mod m$ such that $a^{m-1} \equiv 1 \mod m$ is exactly $$\displaystyle \prod_{p \mid m} \gcd(p-1,m-1).$$ Suppose $f(x) = x^{m−1}−1$ and let $m = ...
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Can it be proved that this extension is algebraic?

Assume that we have a field F, an extension field E of F, and both of them are contained in the algebraic clousure $\overline{F}$. Let E have the property that every automorphism of $\overline{F}$ ...