The extension-field tag has no wiki summary.
9
votes
4answers
108 views
How do extension fields implement $>, <$ comparisons?
I'm taking an abstract algebra course, and we just hit extension fields - for example, you define $\sqrt{2}$ by starting with the field $\mathbb{Q}$ and defining $\sqrt{2}$ as a solution to the ...
8
votes
1answer
105 views
Automorphisms of composite fields of finite extensions
If $\phi:\mathbb{Q}(a_1\ldots,a_n)\rightarrow\mathbb{Q}(b_1,\ldots,b_n)$ is an isomorphism of finite extensions of $\mathbb{Q}$ such that $\phi(a_i)=b_i$, can one extend $\phi$ to an automorphism ...
7
votes
3answers
88 views
Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?
We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
6
votes
2answers
73 views
Equivalent definition of algebraically closed
In Hungerford's Algebra text, it is stated that a field $K$ is algebraically closed iff
there exists a subfield $F$ such that $K$ is algebraic over $F$ and all polynomials in $F[x]$ split in $K[x]$. ...
6
votes
2answers
138 views
degree of a field extension
Let $\alpha$ be a root of $x^3+3x-1$ and $\beta$ be a root of $x^3-x+2$. What is the degree of $\mathbb{Q}(\alpha^2+\beta)$ over $\mathbb{Q}$?
My guess is 9, because i found a monic polynomial of ...
5
votes
2answers
64 views
number of distinct prime Ideal of a ring
could any one tell me how to find number of distinct prime ideals of the ring $$\mathbb{Q}[x]/\langle x^m-1\rangle$$ where $m$ is a positive integer say $4$, or $5$?
what result/results I need to ...
5
votes
1answer
74 views
Field Extensions Problem - From Paolo Aluffi's Book
This is the exercise 4.7, chapter VII, from Paolo Aluffi's algebra book.
I'm sorry for just copying the question without writing any development myself, I don't have a single ideia about how to use ...
4
votes
4answers
174 views
Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$
I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
4
votes
2answers
131 views
$x\otimes 1\neq 1\otimes x$
In Bourbaki, Algèbre 5, section 5, one has $A$ et $B$ two $K$-algebras in an extension $\Omega$ of $K$. It is said that if the morphism $A\otimes_K B\to \Omega$ is injective then $A\cap B=K$. I see ...
4
votes
3answers
91 views
$E/\mathbb F_q$ extension field. Show $[\mathbb F_q(\alpha) : \mathbb F_q]$ is smallest $n$ satisfying property.
Let $q = p^m$. Suppose that $E/\mathbb F_q$ is an extension field and $\alpha \in E$ is algebraic over $\mathbb F_q$. Show that $[\mathbb F_q(\alpha) : \mathbb F_q] = $ the smallest positive integer ...
4
votes
2answers
69 views
Is there a field extension over the real numbers that is not the same as the field of complex numbers?
I'm trying to determine if there is a field, F, such that $\mathbb{R}$ $\subsetneq$ F $\subsetneq$ $\mathbb{C}$ where F is not the same as $\mathbb{R}$ or $\mathbb{C}$.
4
votes
0answers
64 views
Minimal polynomial of a finite purely inseparable field extension
Given $F$, a field with characteristic $p > 0$, and a finite purely inseparable field extension $E$. Then prove that the minimal polynomial $f(x)$ of any $\alpha \in E\backslash F$ will be of the ...
4
votes
0answers
118 views
Proof that a finite separable extension has only finite many intermediate fields
Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
3
votes
2answers
79 views
Why is $X^4 + \overline{2}$ irreducible in $\mathbb{F}_{125}[X]$?
I want to prove that $f = X^4 + \overline{2}$ is irreducible in $\mathbb{F}_{125}[X]$.
I know that $\mathbb{F}_{125}$ is the splitting field of $X^{125} - X$ over $\mathbb{F}_5$, and that this is a ...
3
votes
4answers
126 views
Show $\mathbb{Q}[\sqrt[3]{2}]$ is a field by rationalizing
I need to rationalize $\displaystyle\frac{1}{a+b\sqrt[3]2 + c(\sqrt[3]2)^2}$
I'm given what I need to rationalize it, namely ...
