1
vote
2answers
37 views

Are these subgroups of G only subgroups if G is abelian?

I am doing some exercises in a book I am reading. The exercises and my answers for them are as follows: Let $H$ be a subgroup of $G$, and let $K = \{x \in G: x^2 \in H\}$. Prove that $K$ is a ...
1
vote
2answers
143 views

Algebraic closure exists: What's wrong with this proof?

Given a field $K$, let $U = K[X] \times \mathbb{N}$. Identify each $k\in K$ as $(X-k,1) \in U$, so $K \subseteq U$. Consider fields $(S,+,\cdot)$ where $K \subseteq S \subseteq U$, and the inclusion ...
8
votes
3answers
405 views

Error in proof: $\mathbb{C} \cong \mathbb{C} \times \mathbb{C}$??

I've unintentionally "proved" the following: $$\mathbb{C} \cong \mathbb{C} \times \mathbb{C}$$ Can you help me tracing the error I made resulting to this non-proof? Here it is. First of all, I recall ...
0
votes
1answer
94 views

How Does This Argument Go Wrong?

Consider a binary operation & defined by (a&b)=(a-(b+(a*b))) on the integers, with "a" and "b" as integers. One might argue as follows: By closure (a*b) equals an integer, which we'll call ...
2
votes
1answer
166 views

What is wrong with this proof of Wedderburn's little theorem?

Wedderburn's little theorem $\quad$ every finite domain $A$ is a field. Proof $\quad$ Let $x$ be a nonzero element of $A$. Because $A$ is finite, there exist positive integers $n$, $k$ such that ...
1
vote
1answer
182 views

Erroneous Statement from Linear Algebra

"Theorem": Consider two spaces $W$ and $U$ of finite dimension. If $W \subset U$ then $W^*=U^*$. Proof: $W^\ast$ is a subset of $U^\ast$: $$ \begin{align*} W &= \langle e_1, \ldots, e_k ...