# 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.

56,011 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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