5
votes
Accepted
What is the difference between addition and product in abstract algebra?
This is subtle. You could say that the lesson of abstract algebra is that there is this more fundamental concept of an "operation," and both "addition" and "multiplication&...
- 408k
3
votes
Accepted
Classification of finite groups of order $p^3qr$ up to isomorphism
I suspect there is no complete classification of groups of order $p^3qr$. There is a 2022 paper by Dietrich, Eick and Pan with a complete and detailed classification of group whose orders have at most ...
- 86.8k
3
votes
Voight's Quaternion Algebras, Corollary 7.1.2
Sorry for not seeing this sooner!
There is a major glitch in this argument: if $I \cap I \alpha = \{0\}$, then this does not imply that $I\alpha \subseteq I$! This is just hogwash.
But what is true ...
- 361
2
votes
Does an element of a group to the 0th power equal the identity?
The answer is certainly morally correct. But it might be clearer to say that initially $x^0$ has no meaning, as per group axioms. We must define it. If we define it as $id$ then the original identity ...
- 121
2
votes
Accepted
Mock scalar product's nonlinearity as a derivation
Here is a proof assuming the characteristic is different from $2$. For convenience of typing I will write $\langle u,v\rangle$ as $(u,v)$. Note first that homogeneity with respect to the scalar $s+t$...
- 323k
2
votes
Accepted
Product ideals are the kernel of what ring homomorphism?
I don't believe there is a way to do this. The natural thing to do with two quotients $R \to R/I_1$ and $R \to R/I_2$ is to consider their product $R \to R/I_1 \times R/I_2$, and of course what this ...
- 408k
1
vote
A doubt regarding the extension of Weierstrass factorization theorem
The Weierstraß product theorem is more complicated, it says, that for a set of set zeros and multiplicities ${a_k,n_k}$, without a finite limit point of the postions $a_k$, there exists analytic ...
- 775
1
vote
Applications of Category Theory in Abstract Algebra
There are alot of good reasons to state things categorically.
Universal Properties
Category theory is made to give a precise meaning to universal properties. I don't think it is possible to ...
- 10.2k
1
vote
Accepted
Surjective ringhomomorphism
Let me go line by line.
$\phi(a+b)=\phi((a+b\sqrt{-5})+(c+d\sqrt{-5})) = \overline{a+b+c+d}$
The problem here is that $a+b\neq a+ b\sqrt{-5} + c + d\sqrt{-5}$ in general and what you want to write ...
- 22.3k
1
vote
Accepted
Prime ideals equivalence
It's also easy to see that $1$ or $2$ implies $3$, so we can consider all three equivalent and compare them to the initial definition of prime.
Suppose $A$ is a rng satisfying (any, hence all of) $1,2,...
- 150k
1
vote
Is there a notion of best $\mathbb{K}$-approximations where $\mathbb{K}$ is an algebraic number field?
You can define best approximations in any $\mathbb Q$-vector space, and in particular in any finite field extension of the rational numbers as follows.
Let $K=\{q_0+a_1 q_1 +\dots + a_n q_n:q_i\in\...
- 1,284
1
vote
Is there a notion of best $\mathbb{K}$-approximations where $\mathbb{K}$ is an algebraic number field?
Similar to how we know that the best $\mathbf Q$-approximations of a given real number $x>0$ are given via the partial fractions of its continued fraction representation
It depends on what you ...
- 42.1k
Only top scored, non community-wiki answers of a minimum length are eligible