Tagged Questions
2
votes
1answer
68 views
What is the meaning of $K/F$ is a cyclic extension?
I have it that $K/F$ is a (finite) field extension, what is the definition
of when $K/F$ is called cyclic ?
I heard it while I studied Galois theory and it was defined as
$K/F$ is called cyclic ...
1
vote
1answer
73 views
What is the purpose of the characteristic exponent?
I just came across the term "characteristic exponent" of a field $\Bbbk$. Apparently, it is equal to $1$ if $\DeclareMathOperator{\c}{char}\c(\Bbbk)=0$ and it is equal to $p=\c(\Bbbk)$ otherwise. ...
6
votes
1answer
113 views
An “independence” condition on two algebraic elements over $K$.
Let $K$ be a field and let $a,b\in \overline K$ be algebraic elements.
I've stumbled upon a certain condition on $a,b$, which I feel could be considered an "independence" condition. I would like to ...
2
votes
4answers
249 views
What does it mean to take the splitting field of $f(x)\in F[x]$ over $K$ where $K/F$ is a field extension
Let $K/F$ be a field extension and let $f(x)\in F[x]$. I know $f(x)$ have a splitting field, i.e. a field $E$ that $f(x)$ splits in ($E/F$ and $f(x)$ doesn't split in any proper subfield of $E$).
I ...
4
votes
3answers
224 views
What is the meaning of “algebraically indistinguishable”
I heard the term couple of times (in Field theory class and book),
for example: The different roots of $p(x)=x^3-2$ are "algebraically indistinguishable".
I understand the meaning intuitively, but ...
5
votes
1answer
223 views
Does $\varphi(1)=1$ if $\varphi$ is a field homomorphism?
Is it by definition that $\varphi(1)=1$ if $\varphi$ is a field homomorphism ?
My field theory lecture said that yes, but now $\varphi\equiv0$ is not a field homomorphism...
What is the 'standard' ...
6
votes
1answer
426 views
Why is it called a 'ring', why is it called a 'field'?
The definitions of 'ring' and 'field' are pretty straightforward. For a ring (e.g. integers):
addition is commutative $( 1 + 2 = 2 + 1 )$
addition and multiplication are associative $(2 +(2+2)) = ...
