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Counterexample of if $[F:K]=2$, but $F \supset K$ is not a Galois extension. [duplicate]
I know that if $[F:K]=2$ and the characteristic of $K$ isn't $2$. Then $F \supset K$ is a Galois extension. I think the statement is false, if we drop the condition that the characteristic of $K$ isn'... abstract-algebra group-theory field-theory galois-theory galois-extensions- 5
$\Bbb {Z}_p \cong lim_{←n} \Bbb{Z}_p/p^n \Bbb{Z}_p$(set theoretic bijection) implies $ \Bbb{Z}_p$ is complete as metric space?
$\Bbb {Z}_p \cong lim_{←n} \Bbb{Z}_p/p^n \Bbb{Z}_p$(set theoretic bijection) implies $ \Bbb{Z}_p$ is complete as metric space ? Completeness of ring is often defined as completion as ring $lim_{←n} \... abstract-algebra ring-theory algebraic-number-theory p-adic-number-theory- 29
Can someone explain me what's going on in the Artin-Tate lemma
Let $P\subset Q\subset R$ be rings with $P$ noetherian and $R$ simultaneously finitely generated as a $P$-Algebra and finitely generated as a $Q$ module. Then $Q$ is finitely generated as a $P$ ... abstract-algebra commutative-algebra- 1,083
Quotient of graded ring is graded - confusion about the formalisms
A ring $R$ is graded if it has a direct sum decomposition $R=\bigoplus_{i\in\mathbb{Z}}R_i$ where the $R_i$ are abelian groups and $R_iR_j\subseteq R_{i+j}$. An ideal $I\subseteq R$ is graded if $I=\... abstract-algebra ring-theory graded-rings- 13
Give an example of an ideal I such that $ht(P) < \mu(I)$ for every minimal prime P of I, where $\mu$ is the number of generators
Give an example of an ideal I such that $height(P) < \mu(I)$ for every minimal prime P of I, where $\mu$ is the number of generators. And an example of an ideal I such that $height(P) = \mu(I)$ for ... abstract-algebra commutative-algebra ideals- 11
Existence of maximal ideal in a commutative ring
Let $A$ be a commutative ring, $I \subsetneq A$ a proper ideal of $A$ and $a \in A$ such that $a^k \neq 0$ for all integer $k > 0$. Then there exists an ideal $J$ of $A$ that is maximal satisfying $... abstract-algebra ring-theory maximal-and-prime-ideals nilpotence- 1,030
How do I show this polinomial is irreducible over $\mathbb{Q}$? [duplicate]
I'm trying to show that $P(x) = \frac{x^{p^2}-1}{x-1}=x^{p^2-1}+\cdots +1$ is irreducible over the rationals, as $p$ an odd prime. Here it is my try: As $P$ is a primitive polinomial, it's enough to ... abstract-algebra factoring irreducible-polynomials- 33
Show that $H \cap gK$ is either empty or is equal to the coset of $K \cap H$ in $H$ for subgroups $H,K <G$ and $g \in G$
Problem Statement Suppose $H,K < G$ are subgroups of a group $G$. Prove that for all $g \in G$, $H \cap gK$ is either empty or is equal to a coset of $K \cap H$ in $H$. A quick question to get ... abstract-algebra group-theory- 1,431
Normalizer of a subgroup in $\mathbb{Z}_2^{\otimes 2n}$
Does anyone know a fast method/algorithm for calculating the normalizer of an abelian subgroup $G$ of $ \mathbb{Z}_2^{\otimes 2n}$ (equipped with a symplectic inner product)? Or do I need to check ... linear-algebra abstract-algebra- 1
Show That Wigner’s Theorem Defines a Homomorphism $\operatorname{U}(2) \rightarrow \operatorname{SO}(3)$
Preliminary Knowledge We are working on the finite dimensional Hilbert space $\mathbb{C}^2$. The projective Hilbert space is given by $$\mathbb{P}(\mathbb{C}^2)=\big(\mathbb{C}^2 \backslash \{0\}\big)... abstract-algebra group-theory group-homomorphism quantum-mechanics- 232
Non existence of a polynomial between two vectorspaces
Let be $V$ the vector space of the sequences in $\mathbb{C}$ and $\varphi: V \rightarrow V,(x_1,x_2,...) \mapsto (x_2,x_3,...)$ Show that there is no polynomial such that $f \in \mathbb{C}[x] \... linear-algebra abstract-algebra- 1,121
Is there a video playlist or videos of lecture of graduate level abstract algebra?
Is there a video playlist or videos of lecture of graduate level abstract algebra? I'll be taking a graduate level algebra class this coming fall and will be using Abstract Algebra by Dummit and Foote.... abstract-algebra- 29
If $g$ is in group $G$, then $g$ belongs to a subgroup $H$ only if $gH = H$
I have tried to prove this but I am unable to proceed. I have been successful in proving that $gH = H \implies g \in H$ but I am unable to prove the reverse. abstract-algebra group-theory- 21
Proving free group $F_2$ is not isomorphic to $F_3$ [duplicate]
I'm trying to prove the free group on 3 generators is not isomorphic to the free group on 2 generators. I have that there are many injections $F_3 \hookrightarrow F_2$ and of course $F_2 \... abstract-algebra group-theory free-groups- 204
The quotient group $(\mathbb{R}\times \mathbb{R},+)/\{(a+b\sqrt{2},a-b\sqrt{2}):a,b\in\mathbb{Z}\}$
As said in the title, I'm trying to find a representation of the quotient group $(\mathbb{R}\times \mathbb{R},+)/ \{ (a+b\sqrt{2},a-b\sqrt{2}):a,b\in\mathbb{Z} \}$ by finding a homomorphism $f$ on $\... abstract-algebra group-theory representation-theory quotient-group- 696
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