# Tagged Questions

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### Determining final and initial object in a certain category

I am reading Paolo Aluffi's greatly entertaining book "Algebra: Chapter $0$" and I got stuck on some excercises dealing with universal properties. Let $C$ be a category, and let $A$ and $B$ be two ...
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### Introductory texts in abstract algebra, and game theory taking non-standard approaches

I like to see subjects from different angles. For example in linear algebra I'm reading through Axler's text (which takes a proof based approach for math students), but I'm also checking out a text on ...
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### If $G$ is a simple $f$ an homomorphism, and $A\lhd H$ is such that $[H:A]=2$, show $f(G) \subset A$

I'm stuck with the following problem. Can someone help me by providing a hint? Suppose $G$ is simple and let $f$ be an homomorphism between $G \to H$. If $\#G\ne2$, $A\lhd H$, and $[H:A]=2$. Then ...
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### hint with an exercise algebra

I'm stuck with the following I hope someone could help me. Let $N$ a normal subgroup of $G$. Show that if $[G:N]=4$, exists a normal subgroup $M$ of $G$ s.t. $[G:M]=2$. My idea: Since $G/N$ has ...
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### How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: ...
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### Introduction and Prerequisites to Abstract Algebra

So I've seen similar questions asked, but none that really helped me out. I'm going to be a freshman in college next year, having already taken Multivariate Calculus and Elementary Linear Algebra. Of ...
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### Find the galois group of the polynomial when a root is given

If $\alpha$ is a root of a polynomial $f(x)=x^3 +x^2-4x+1$ then show that $2 - 2\alpha - \alpha^2$ is also a root of $f(x)$. Use this fact to compute the Galois group of the splitting field of $f(x)$ ...
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### Finding a cycle with a specific property

I am reading the book Dummit and Foote - Abstract Algebra . One of the exercises is to find an $n$-cycle $(n \ge 5)$, $\sigma$ such that $\sigma^k = \tau$ for some positive integer $k$, where ...
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### Abstract Algebra: prove it is cyclic

I have question in referring to below link. Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic? I ...
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### Normalizer of a subgroup of $GL_2(\mathbb{R})$

I have following subgroup of $GL_2(\mathbb{R}):$ A=\Bigg\{\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right),\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} ...
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### Is there any motivation for the direct sum?

I believe that everything have reason to exist . I want to learn why the direct sum exist. Is there any motivation for the direct sum to exist ?
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### The form of subrings of $k[[t]]$

I saw this question in an algebraic geometry book. I tried to solve this. But I did trivial thing, so I don't write what I did here. This is just self-studying. I want to learn how to solve. Please ...
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### Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
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### Free $A$-module isomorphic to a direct sum of copies of $A$?

Does this proposition hold even if it's not finitely generated? I think it does, since $M$ isomorphic to the direct sum of $M_i$, $M_i$ isomorphic to the direct sum of ...
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### Abstract Algebra: Irreducible polynomials and automorphism

While I was studying and referring few examples I came up with few question. Hope you guys can help me solving my confusion. Irreducible polynomials problem 1 in this how to argue that why the ...
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### The meaning of a symbol in the proposition

The question is what's the meaning of the symbol $\phi$? If it just a mapping,what's the mean of the equation? I guess it's $\phi(x)$, $x$ is the element of $M$. Then $\phi(x)$ is the element of ...
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$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
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### Finite field and Automorphism

Problem 1. Let S be a finite field of characteristics 2 and the map be define as $\eta$: S$\longrightarrow$S x$\longmapsto$x$^p$ Show that $\eta$ is automorphism, i.e., S is isomorphism ...
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### Question of a proposition about direct product

I try to prove it's injective, surjective and homomorphism. define f(x)=(x+a1,x+a2,....,x+an),it's homomorphism. it's injective <=> the intersection of ai=0 I don't know how to prove the ...
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### proposition 1.10 ii) A&M Introduction of commutative algebra

I am working through Introduction of commutative algebra and am having trouble with the following question: (I'll use f instead of the map,since I don't know how to input it.) Q1: Why there exist ...
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### is there a counterexample of this map isn't surjective?

The ring A is a commutative ring with identity. I think ii) is true if they are not coprime. because for every (x+a1,....,x+an) we can find a x such that f(x)= (x+a1,....,x+an). Could you please ...
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### Smallest Algebra Containing Singletons

$\Omega:=\mathbb N$. What is the smallest algebra containing all singleton $\{\omega\}$, i.e. $\{1\}, \{2\}$, and so on. Any hint, please?
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### Don't understand a proposition and its proof

Proposition 5.1 from Commutative Algebra by Atiyah and Macdonald: $x∈B$ is integral over $A$,then $A[x]$ is a finitely generated $A$-module. The elements in $A[x]$ are the set of all the sum. If ...
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### Abstract Algebra: Ring Homomorphism injective

Reference: Ring Homomorphism $\phi:f\to S$ is injective Referring this I have a doubt which I needed to clear. Below is my answer and query. We know $\phi:f\to S$ be ring homomorphism, where $f$ is a ...
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### What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that:
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### Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
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### Are different constructions of an algebraic structure always isomorphic?

Any two complete ordered fields are isomorphic (as proved, e.g., in Spivak's Calculus; see also this question). While I understand this proof, I cannot yet appreciate why it is necessary. Given any ...
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### Is it possible to learn abstract algebra no precalculus or calculus?

See above. I am trying to re-teach myself mathematics in a different manner than is formally taught (i.e., set theory, number theory, mathematical logic, abstract algebra, discrete math and then ...
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### Could you please explain the detail of the proof

Proposition: Proof: Question: Why it's isomorphism?
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I even don't know how to start.
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### primitive root of residue modulo p

I was trying to prove that for the set $\{1,2,....,p-1\}$ modulo p there are exactly $\phi(p-1)$ generators.Here p is prime.Also the operation is multiplication. My Try: So I first assumed that if ...
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### Number of groups of a given order

In general, for what $n$ do there exist two groups of order $n$? How about three groups of order $n$? I know that if $n$ is prime, there only exists one group of order $n$, by Lagrange's Theorem, but ...
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### How to calculate the tensor product?

This question might be stupid for you.I ask because I have no clue about it. I don't really understand what is tensor product,although I know its definition. I have search what is tensor product,so I ...
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### Why these two propositions have different requirements

Proposition 2.18 is similar to 2.19. Why we need $N$ flat in 2.19? What's the difference between 2.18 and 2.19?
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### Affine variety example.

I know how to show the statement. But I cannot find an example (the part I underlined by a yellow pen) please help me for finding an example. Note: For example, can I consider the following ...
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### Why we cannot in general define the product of two submodules

why we can define product of two ideals of a commutative ring,but we can't in general define the product of two submodules? ...
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### How to prove M/Ker(f) & Im(f) are isomomorphic

f:M->N us an A-module homomorphism. A is a commutative ring. How to prove M/Ker(f) and Im(f) are isomomorphic I can't prove this statement. But if A is a field,it's not hard to be proved. For ...
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### Showing if $H,K$ are subgroups of $G$, the $\varphi(hak)=a^{-1}hak$ is bijective.

Here's my problem...I have to ultimately show that $\varphi(hak)=\varphi(hbk)\Rightarrow hak=hbk$ and I'm having issues... I am told, both $H,K$ are subgroups of $G$ and $a\in G$. The goal is to show ...
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### Question about construction of an algebraic closure

$A=K[x]$,$\mathfrak{m}$ is a maximal ideal containing a principle ideal of $A$. every element of $K[x]/\mathfrak{m}$ can be described by form $f+\mathfrak{m}$. every element of $K$ is the ...
In Constructing algebraic closures by Keith Conrad, the author writes: Let $K$ be a field. We want to construct an algebraic closure of $K$, i.e., an algebraic extension of $K$ which is ...