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Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify which topic of abstract algebra is most related to your question and also to help other users find similar questions through the search engine.

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Hatcher Exercise 3.2.9

Show that if $H_n(X; \mathbb{Z})$ is free for each $n$, then $H^∗(X; \mathbb{Z}_p)$ and $H^∗(X; \mathbb{Z})⊗\mathbb{Z}_p$ are isomorphic as rings. I'm assuming the tensor product is taken over $\...
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On the kernel of the map $k[X,Y]\to k[T],X\mapsto T, Y\mapsto T$

The following exercise, with a slightly more interesting choice of $\varphi$, is probably a standard exercise in, say, an introductory course in commutative algebra. Let $k$ be a field. Show that ...
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Let $k$ a field, $k'$ a subfield, and $A$ any associative $k$-algebra. Can a quotient of $A$ ever yield $k'$?

I am trying to learn some basic Scheme theory out of Eisenbud's book "Schemes: the Language of Modern Algebraic Geometry." I'm trying to understand how elements of a ring can be treated as functions ...
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2answers
47 views

Prove that a primitive $q$-th root of unity is in the algebraic closure of $\Bbb F_p$

Let $p$ and $q$ be odd primes. Let $\Omega$ be the algebraic closure of $\Bbb F_p$. Let $\omega$ be a primitive $q$-th root of unity. Show that $\omega \in \Omega$. How do I show that? Please help me ...
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Question about Semigroups, Monoids, and Groups

My book defines semigroups, monoids and groups in a traditional way. Where semigroups are defined with associativity, monoids with associativity and identity and groups with associativity, identity ...
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31 views

Transcendental Extension and Algebraic Extension commute.

I want to show the following: Let $L|K$ be an algebraic field extension of $K$. Let $T$ be transcendental over $K$. Then $$ K(L)(T) = K(T)(L). $$ We defined the adjunction $K(A)$ of a subset $A \...
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1answer
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Finding square roots of quadratic residues in prime power field

I know that in fields of cardinality $p$, $a$ is a quadratic residue if and only if $a^{\frac{p-1}{2}}=1$ (Euler's criterion). Therefore $a^{\frac{p+1}{2}}=a$ and if also $p=3\!\!\!\mod\! 4$ we can ...
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+50

If $K/F$ is normal then $K/I$ is Galois.

Let $K/F$ be a normal extension and $I$ be the inseparable closure of $F$ in $K.$ Let $G=\text{Aut}_F(K),$ i.e., $F$ isomorphisms on $K$and similarly define $H=\text{Aut}_I(K).$ Now I have already ...
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2answers
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Irreducible elements of the ring of all numbers of the form $2^ab,$ where $a$ and $b$ are integers

As the title explains, I'm trying to solve a question which asks me to determine which are the irreducible elements of the ring of numbers of the form $2^ab,$ where $a$ and $b$ are integers (with the ...
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1answer
69 views

Cayley-Hamilton Theorem proof

I was asked to deduce the Cayley-Hamilton Theorem, that is to show that for all $A \in M_{n}(\mathbb{C})$ we have $\chi_{A}(A)=0$, from the following Corollary: Let $A$ be an $n \times n$ matrix over ...
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How $H \cap P$ is a Sylow $p$-subgroup of $H$?

I recently came across the following problem : True or false: If $P$ is a Sylow $p$-subgroup of a finite group $G$, then for any subgroup $H$ of $G$, $H \cap P$ is a Sylow $p$-subgroup of $H$. The ...
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1answer
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Determine $[\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11}):\mathbb{Q}]$, and determine if $\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11})/\mathbb{Q}$ is normal

Determine $[\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11}):\mathbb{Q}]$, and determine if $\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11})/\mathbb{Q}$ is normal. If we let $x = \sqrt[4]{11}+i\sqrt[4]{11} = \sqrt[4]{...
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1answer
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Find the splitting field $K$ of $x^{12}-9$ over $\mathbb{Q}$ and determine $[K:\mathbb{Q}]$.

Find the splitting field $K$ of $x^{12}-9$ over $\mathbb{Q}$ and determine $[K:\mathbb{Q}]$. My approach: First we can factor it $x^{12}-9 = (x^6-3)(x^6+3)$ so that the first factor gives us that $\...
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1answer
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Question on Proof of ismorphism between product and coproduct under Abelian group

In Steve Awodey's book Category Theory, he has some proof that seems wrong to me on page $60$, he mentioned a proposition that Proposition $3.11.$In the category $Ab$ of abelian groups, there is a ...
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Grassmannians as Gelfand Pairs

Why are $(O(n), O(k) \times O(n-k))$ and $(U(n), U(k) \times U(n-k))$ (corresponding to the real and complex Grassmann manifolds) symmetric Gelfand Pairs? Is this true for $(Sp(n), Sp(k) \times Sp(n-...
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1answer
79 views

finitely generated projective module and Nakayama's lemma

Let $R$ be a local ring with maximal ideal $I$. $M$ is a finitely generated module over $R$ generated by $a_1, \ldots, a_n$ and the generators are chosen such that their quotients in $M/IM$ form a ...
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2answers
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Let $f : G → G_1$ be a surjective homomorphism (also called epimorphism) from $G$ to another group $G_1$. Prove that $f(Z(G)) \subseteq Z(G_1)$.

I am working on a school assignment and have been stuck on this question for some time. Let $f : G \rightarrow G_1$ be a surjective homomorphism (also called epimorphism) from $G$ to another group $...
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3answers
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Does $x^3 \equiv 6 \pmod {11}$ have any solutions?

My gut instinct is that it does not, but I am unsure of how to show this... and I think the $x^3$ is what is causing me trouble in figuring this out. I have attempted to rewrite it using the ...
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1answer
44 views

Why is the multiplication of permutations not commutative?

If the multiplication of disjoint cycles is commutative, and every permutation can be written as the product of disjoint cycles, then why is the multiplication of permutations not commutative?
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3answers
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All of the prime numbers s.t. $3x^p+px^3\equiv \,\,0\,\, (\text{mod} \,\, 5)$ has solutions $\forall x \in \mathbb{Z}$

Determine all of the prime numbers $\,\,p\,\,\,$ s.t. $\,\,\,2 \le p \le 50\,\,\,$s.t. the following congruence has solutions $\forall x \in \mathbb{Z}$ $$3x^p+px^3\equiv \,\,0\,\, (\text{mod} \,\, 5)...
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1answer
69 views

$\mathbb{Z}$-Module exercise

I am trying to solve the following exercise on basic module theory and I am stuck. Any help would be more than welcome! So let $M\subseteq \mathbb{Z}^3$ the solutions to the following problem: $-3x+...
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Ideals in ring of polynomials.

Let $ \ I$ be an ideal in $\ k[x_1,...,x_n]$. 1) Prove that $\ 1 \in I$ if and only if $\ I =k[x_1,...,x_n]$. 2) Prove that $\ I$ contains a nonzero constant if and only if $\ I =k[x_1,...,x_n]$. ...
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0answers
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If $G = H \cup K \cup L$, then $[G:H] = [G:K] = [G:L]=2$ [duplicate]

This is a problem from Isaacs's Algebra. Let $G$ be a finite group and $H, K, L$ be proper subgroups of $G$ such that $G= H \cup K \cup L$. He asks us to show that, in this case, $[G:H] = [G:K] = [G:L]...
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1answer
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How to show a piecewise binary operation over functions is associative?

Given some set of functions $S(X) = \{f:[0,a]\to X : f(0)=f(a)=x_0 , a \geq 0\}$, define the binary operator $*$ over two functions $f:[0,a]\to X$ and $g:[0,b]\to X$ as: $$(f * g)(t) = \begin{cases} f(...
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How to show that a given polynomial is irreducible in a cyclotomic field

I'm beginning to study McCarthy's Algebraic Extensions of Fields, and one of the first problems is to give a factorization of $x^4 + 1$ in $K[x]$, where $K=\mathbb{Q}(a)$ and $a$ is a root of $x^4+1$ (...
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1answer
46 views

Finitely generated subgroup of FINITE order of a group of homeomorphisms of closed unit disc

Example of a non-trivial finitely generated subgroup of finite order of a group of homeomorphisms (which fix boundary point wise) of closed unit disc? This question is related to this one (Finitely ...
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1answer
58 views

What is the order of the subgroup $\langle 5\rangle \times \langle 3\rangle$ in $Z_{30} \times Z_{12} ?$

What is the order of the subgroup $\langle 5\rangle \times \langle 3\rangle$ in $Z_{30} \times Z_{12} ?$ I think it should be $12$, since $O(5) = 6$ in $Z_{30}$ and $O(3) = 4$ in $Z_{12}$. The order ...
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2answers
21 views

Finitely generated subgroup of infinite order of a group of homeomorphisms of closed unit disc

Can we construct a finitely generated subgroup of infinite order of a group of homeomorphisms (which fix boundary point wise) of closed unit disc?
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0answers
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Proof of “Lagrange's Lemma”

Can anyone share a link of proof of the following fact ? Let $f(x) \in K[x] $ be an irreducible polynomial with $n$ distinct roots $r_i$ and let $g(x_1,\dots, x_n)$ and $h(x_1,\dots, x_n)$ be ...
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1answer
44 views

Show that this holds in every Boolean Algebra

Prove that this holds in every boolean algebra: $$ x\land y \land z' =x \space \space \text{iff}\space \space x \lor y =y \space \space and \space \space x\land z=0 $$ My guess is to start first with ...
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1answer
61 views

symmetric functions vs symmetric polynomials

I am doing my thesis related with symmetric functions and representations. It is for this reason that I am reading MacDonald's book Symmetric Functions and Hall Polynomials. When reading chapter 1....
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2answers
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How to show that $G=\{a+b\sqrt 2\mid a,b\in \mathbb{Z} \}$ is not a cyclic group?

Show that $G=\{a+b\sqrt2\mid a,b\in\mathbb{Z}\}$ is not cyclic under addition. Trying to show that $G$ is not a group, and without showing that $G$ is isomorphic to $\Bbb{Z}\times\Bbb{Z}$, it seems ...
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0answers
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Can we quotient a set $A$ by another set $B$ such that $B\not\subset A$?

I'm reading Ash's Basic Abstract Algebra. Here: I'm a little bit confused. When we restrict the map $\pi: G\to G/N$, we get the map $\pi_0:H\to H/(H\cap N)$, right? Assuming this is true, I don't ...
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Proving $(U \cap V) \cap u(U \cap v) = (u \cap V)(U \cap v)$

Let $U, V$ be subgroups of a group. Let $u \trianglelefteq U, v \trianglelefteq V$. I proved like this. Then by applying modular law, $$\mathrm{(LHS)} = (U \cap V) \cap u(U \cap v) = U \cap V \cap uv \...
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2answers
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Confusion in notation, problem from Herstein's book.

Let $G$ be a nonempty set closed under an associative product, which in addition satisfies : A. There exists an $e$ in G such that $a⋅e=a$ for all $a∈G$. B. Given $a∈G$, there exists an ...
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1answer
79 views

Axes of a free module over a PID

Let $R^2 = R \times R$ be a free module of rank $2$ over a principal ideal domain. I am trying to prove that for every non-zero element $a$ of $R^2$ there is a basis such that $a$ belongs to one of ...
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1answer
25 views

Extension of automorphism of field

Let $F$ be a field of characteristic zero, $\overline{F}$ be the algebraic closure of $F$. Let $\zeta_n$ be a primitive $n$-th root of unity in $\overline{F}$. Then it is well-known that $F(\zeta_n)$ ...
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0answers
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Elements of $\mathbb{Q}(\zeta_p)$ fixed under $\zeta_p \mapsto \zeta_p^g$

I'm reading Harold Edwards’ Galois theory and before going into Galois theory he was discussing about Gauss' works on constructible $n$ gons (so please avoid Galois Theory while answering this ...
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1answer
32 views

The size of the stabilizer of the nth power of the Frobenius Map on a field

Fix a prime $p$. Let $F$ be a field of order $p^k$ and with characteristic $p$. Let $\phi$ be the Frobenius map, which maps $a$ to $a^p$; one can check that, for a field of characteristic $p$, it is a ...
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1answer
65 views

So many different 'varieties', which one is this? Serre's algebraic variety

Anyone who has ever tried to study algebraic geometry has experienced the phenomenon of being burdened by countless types of varieties (variety, affine variety, projective variety, quasi-affine ...
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1answer
44 views

Do the roots of a polynomial remain inside the unit circle if we make the coefficients all positive?

Suppose the roots $r_1, \dots, r_p$ of the polynomial $$x^p + a_1 x^{p-1} + a_2 x^{p-2} + \dots + a_p$$ all lie inside the unit circle. Is it true that the the roots of the polynomial $$x^p + |a_1| ...
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1answer
64 views

Show that $Ax=0$.

I need a hint to help me get started with this problem: Given the sequence of homomorphisms, $\mathbb{Z}^{m}\to \mathbb{Z}^{n} \to M\to 0$, where $M=\mathbb{Z}^{n}/K$ and $K=im(\phi_{A})\subseteq \...
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0answers
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Is two subsequent Kummer extensions again a Kummer one?

Let $F$ be a field of characteristics $p$ such that $\mu_n$, $\mu_m$, $\mu_{nm} \subset F^*$ for $p \nmid n, m \in \mathbb{N}$. Consider two subsequent Kummer extensions $F \subset K \subset L$, that ...
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1answer
39 views

Can I multiply an element of a group to an element of a different group?

The question goes like this: Write down a group table for the groups C4 and C2 x C2. For every element a in C4 and C2 x C2 determine the smallest positive integer m such that $ m\cdot a$ equals the ...
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1answer
32 views

Explicit maps behind the isomorphism $k \otimes V \cong V$

This question is undoubtably extremely trivial, especially since every text I read gives only the forward direction map behind the isomorphism $k\otimes V \cong V$, i.e. $\lambda \otimes v \mapsto \...
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2answers
35 views

What algebraic structure encapsulates multiplication of a vector by a matrix?

$\newcommand{\R}{\mathbb{R}}$ tl;dr: In what way is $w = T(v)$ "the same" as $[w]_C = {}_C [T]_B [v]_B$? Here $V,W$ are vector spaces, $B,C$ are respective bases for them, and $v \in V$, $w \in W$, $T ...
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0answers
35 views

Storing numbers in an efficient way in computer.

I know that we can write a very large number such as 5040 in only 7!, and imagine I want to store this number in a binary file with the least number of bits. Saving 5040 takes 13 bits of space, while ...
2
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1answer
31 views

Every finite field has an extension in which every element has a square root

I know every field has an algebraic closure, I just wanted to prove this statement separately using "simpler" methods. Let $(K,+,\cdot)$ be a finite field. First suppose there exists an $x \in K$ ...
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0answers
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Prove the structure constant of the class algebra satisfies $f(c,d,e) = f(\bar{c},\bar{d},\bar{e})$

This is problem 4 in A. Zee's book, Group Theory in a Nutshell for Physicists. Consider an equivalence class $c=\{ g_1^{(c)},...,g_{n_c}^{(c)}\}$, define the class average $$K(c) = \frac{1}{n_c}\sum_{...
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2answers
36 views

Showing that $(x+1)$ is prime but not maximal in $\mathbb{C} [x,y]$

I'm trying to show that $(x+1)$ is prime but not maximal in $\mathbb{C} [x,y]$. Is is not maximal because $(x+1)\subset (x)$. But why is it prime? I know that it is enough to show: x + 1 is prime, ...