0
votes
0answers
18 views

Notational issue

Let $K = F(t)$. If $r \in K: (\nexists c \in F: r(t) = c \forall t)$ is a rational function and $L = F(r(t))$, then what form does $f \in L$ have? Is it a rational function where the coefficients are ...
0
votes
2answers
39 views

If two elements a and b in a field extension L/K are algebraic in K and have the same minimal polynomial, prove that $K(a)=K(b)$

I have a field extension $L/K$ and two elements $a,b \in K$, which are algebraic over $K$ and both have the same minimal polynomial, and need to prove that then $K(a)=K(b)$. I can see how this can be ...
3
votes
2answers
148 views

Prove that $\mathbb Q(\sqrt 2) \neq \mathbb Q(\sqrt 3)$

I've tried writing out the contents of each and attempting to get a contradiction by equating arbitrary elements but can't get this to work. I can't think of any counterexamples as everything I come ...
0
votes
0answers
38 views

Given $K = \mathbb Q[x]/(x^2 - 5)$, a field extension of $\mathbb Q$, prove that $K \simeq \mathbb Q(\sqrt 5)$

My reasoning so far is that $(x^2 - 5) \cong 0$ in K, so we can set $x^2-5=0$ and get $x=\pm \sqrt 5$, which to me suggests $K \simeq \mathbb Q[x]$ but I'm having trouble jumping to the field of ...
3
votes
0answers
73 views

Question about Wantzel's proof of the necessary condition for compass/straightedge constructibility

I'm trying to understand Wantzel's original proof of the necessary condition for constructibility with a straightedge and compass. It's expressed in terms of polynomials rather than field extensions. ...
1
vote
0answers
54 views

Subfields $k\subseteq F\subseteq k(x_1,\dots,x_n)$ and their intersection with $k[x_1,\dots,x_n]$

By this thread, if I have a subfield $k\subseteq F\subseteq k(x_1,\dots,x_n)$, $F$ is of the form $F=k(\phi_1,\dots,\phi_m)$ for some rational functions $\phi_1,\dots,\phi_m\in k(x_1,\dots,x_n)$. But ...
1
vote
1answer
137 views

minimal polynomial of $x$ over $ K\left(\frac{p(x)}{q(x)}\right) \subset K(x) $

Let $K$ be a field , let's consider the field of rationals functions over x , $k(x)$. Let $t\in k(x)$ be the rational function $\frac{p(x)}{q(x)}$ , where $P,Q$ have no common factors. I have to prove ...
4
votes
1answer
232 views

What is the Galois group of the extension $\mathbb F_3(x^4)\subset\mathbb F_{3^2}(x)?$

Square brackets $[\;]$ will denote taking the ring of polynomials, and round brackets $(\;)$ will denote taking the field of rational functions. My homework assignment from about a month ago had the ...
4
votes
2answers
243 views

When a field extension $E\subset F$ has degree $n$, can I find the degree of the extension $E(x)\subset F(x)?$

This is not a problem I've found stated anywhere, so I'm not sure how much generality I should assume. I will try to ask my question in such a way that answers on different levels of generality could ...
0
votes
1answer
481 views

The field of rational functions in $n$ variables is a Galois extension of the field of symmetric rational polynomials in $n$ variables.

I've been doing a little bit of field theory for number fields but not much with function fields. The question originally asked says "For some field F, show that the field $F(u_1,\ldots, u_n)$ is a ...
10
votes
2answers
943 views

Luroth's Theorem

I have just begun to read Shafarevich's Basic Algebraic Geometry. In the first section of the first chapter, he quotes Luroth's theorem, which states that any subfield of $k(x)$ that is not just $k$ ...