Use this tag for questions about extension fields in abstract algebra. An extension field of a field K is just a field containing K as subfield, but interesting questions arise with them. Use this topic for dimension of extension fields, algebraic closure, algebraic/transcendental extensions, ...

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5
votes
1answer
64 views

Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$

I don't think I fully understand. Let's say there is a root $x_0 \in K=\mathbb{F}_p(t)$, where $p$ is a prime number. Then $x_0 = \frac{P(t)}{Q(t)}$ for some polynomials $P,Q \in \mathbb{F}_p[t]$. ...
1
vote
1answer
27 views

Proof of a result on closed subgroups of Galois group

Let $M\supseteq K$ be an algebraic normal extension. Then its Galois group is profinite. I have been told that in this hypothesis, if $H\leq G$, then $H''=H\iff \bar H=H$, where ...
1
vote
1answer
25 views

Show that a field extension $L/K$ is separable iff the trace form is non-degenerate.

Let $L$ be a finite field extension of $K$. I have the following question: Show that $L/K$ is separable if and only if the bilinear trace form $\text{Tr}_{L/K}:L\times L \to K$ is non-degenerate. ...
3
votes
1answer
69 views

Matrix and field extension

It is given that $F\subset K$ are fields. $A$ is a matrix of size $n\times n$ over $K$. I need to prove that there exist $c_1,\ldots,c_k\in K$, linearly independent over $F$, and matrices ...
4
votes
2answers
41 views

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group with some elementary algebra, $x - \sqrt{2 +\sqrt{2}} = 0 ...
0
votes
1answer
26 views

Radical extension with root of cubic polynomial

If I take $f(x)$ is an irreducible cubic over $\mathbb{Q}$ with a root $\alpha$ in a splitting field and given that $\mathbb{Q}(\alpha)$ is a radical extension is it true that $\mathbb{Q}(\alpha) = ...
0
votes
1answer
41 views

Prove that $L=K[x]/\langle m(x)\rangle$ is an algebraic extension of $K$

Let $m(x)$ be an irreducible polynomial of degree $n$. How can I prove that $$L=\frac{K[x]}{\langle m(x)\rangle}$$ is: A vector space of dimension $n$, An algebraic extension of $K$? Edit: I'm ...
3
votes
1answer
57 views

Fields of characteristic $p$ that are isomorphic as vector spaces, but not as fields

Give an example of a field $\mathbb{K}$ of characteristic $p > 0$ and elements $a,b \in \bar{\mathbb{K}}$, such that $\mathbb{K}(a)$ and $\mathbb{K}(b)$ are isomorphic as vector spaces over ...
4
votes
0answers
37 views

extending isomorphism to an automorphism

Lets say $K/F$ is a field extension and $\alpha ,\alpha '\in K$ are two distinct roots of the same irreducible polynomial in $F[x]$. there exists an isomorphism $$\psi:F(\alpha)\rightarrow ...
1
vote
3answers
30 views

What is the number of elements in $Aut(Q(\pi)/Q)$?

I tried to prove that $|Aut(E/F)|$ is finite, then $E/F$ is a finite extension, but then now I think $Q(\pi)/Q$ would be a counterexample for this. I can see that there are two automorphisms ...
1
vote
1answer
51 views

Show every $a \in E^*$ is a root of $x^{p^d-1} -1 $?

Let $\mathbb{Z}_p < E$ be an extension field of degree $d$. A simple counting argument shows: $|E^*| = p^d - 1$ Proposition: For all $\alpha \in E^*$, $x^{p^d-1} -1 = 0.$ In a field of $p^d ...
1
vote
1answer
20 views

Give an extension field of $\mathbb{Z}_3$ of degree 3?

I have an irreducible polynomial in $\mathbb{Z} $ That irreduible polynomial is: $(1)$ $x^3 + 2x + 1$ I know that this polynomial creates a maximal ideal and that I can create an extension field ...
0
votes
3answers
57 views

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$ Let $\sigma$ be such that $\sigma(t)=-t$. I assume there is only one automorphism like this, I am not sure exactly why... How ...
1
vote
1answer
40 views

Show that $|\operatorname{Aut}(\mathbb{Q}(\sqrt[10]2))|=2$

Let $\mathbb{K}=\mathbb{Q}(\sqrt[10]2)$. Show that $|\operatorname{Aut}(\mathbb{Q}(\sqrt[10]2))|=2$. Well, it is easy to see that the degree of this extension over $\mathbb{Q}$ is ten. Also, is ...
6
votes
0answers
56 views

Let $\mathbb{K} $ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K} $ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then ...
2
votes
0answers
22 views

How to prove that any infinite algebraic extension of a complete field is never complete?

My first idea is using Baire category theorem since I thought an infinite algebraic extension should be of countable degree. However, this is wrong, according to this post. This approach may still ...
0
votes
4answers
73 views

Consider extension $[\mathbb{Q}(\alpha\ ):\mathbb{Q}]$ where $\alpha\ $ is zero of $P(x) = x^4 + 9x^{2} + 15 $.

Consider extension $[\mathbb{Q}(\alpha):\mathbb{Q}]$ where $\alpha$ is zero of $p(x) = x^4 + 9x^{2} + 15 $. Find $[\mathbb{Q}(\alpha):\mathbb{Q}(\alpha^2 + 3)]$. My attempt: By Eisenstein's ...
2
votes
2answers
49 views

Is the degree of an infinite algebraic extensions always countable?

I guess this is right and try to prove it by using the fact that the polynomial ring $K[t]$ has a countable basis $1,x,x^2,\cdots$. But How to use this fact? Aside, if this statement is true. Is the ...
1
vote
1answer
26 views

Field Theory Problem in Beachy's Abstract Algebra involving field extensions and transcendental elements.

Let $\mathbb{F}=\mathbb{K}[u]$ where $u$ is transcendental over $\mathbb{K}$. Show that if $\mathbb{K} \subsetneq \mathbb{E} \subseteq \mathbb{F}$ then $u$ is algebraic over $E$. I'm guessing ...
1
vote
1answer
20 views

Let $K\subseteq F$ be a finite extension of fields, $M$ and $N$ intermediate fields separable on $K$. Show that $MN$ (compositum) is separable.

Let $K\subseteq F$ be a finite extension of fields, $M$ and $N$ intermediate fields separable on $K$. Show that $MN$ (compositum) is separable. Using the Element Primitive Theorem, we know that ...
2
votes
1answer
45 views

Extension of fields $[L:K]=2$. Show that there is a element $\alpha\in L$ such that $L=K(\alpha)$ and $\alpha^2\in K$.

Let $K$ be a field with characteristic different from two, then if a extension of fields $L/K$ is such that $[L:K]=2$ then there is a element $\alpha\in L$ such that $L=K(\alpha)$ and $\alpha^2\in K$. ...
1
vote
1answer
26 views

$F(t)$ as an $F[t]$-algebra and the Weak Nullstellensatz

Sorry if this question has already been answered somewhere, but it's quite hard to find if so, because of the use of the word 'algebra' in the question... In the lead up to a proof of the Weak ...
1
vote
1answer
24 views

Roots of $X^{l-1}+1$ in a quadratic extension $F_q$, $q=l^2$

Consider a finite field $F_q$ where $q=l^2$ ($l$ can be of the form $p^m$). Does $F_q$ has a root of $X^{l-1}+1$? As $X^l+X = X(X^{l-1}+1)$ we can show that $X^{l-1}+1$ splits if it has a root. This ...
1
vote
2answers
32 views

Prove that $p(x)$ is irreducible in $F[x]$

Let $F$ be a field and let $K$ be an extension of $F$. Let $\alpha$ be algebraic over $F$. Let $p(x)$ be the polynomial of minimal degree having $\alpha$ as a root. Prove that $p(x)$ is irreducible in ...
1
vote
2answers
46 views

Field extension over $\mathbb{Q}$

I am given a subfield $E$ of $\mathbb{C}$ and asked to show that $[E : \mathbb{Q}] \le 10$ when every element of $E$ is a root of a polynomial in $\mathbb{Q}[x]$ of degree $10$. But I don't think ...
3
votes
0answers
32 views

Prove the form of a fixed field

let $C_3=\langle\sigma\rangle$ and let $\sigma$ act on $K(s,t)$ by $s \mapsto t$ and $t \mapsto -s-t$ I want to prove that $K(s,t)^{C_3}=K(u,v)$ with $u=\frac{s^2+t^2+st}{st(s+t)}$ and ...
0
votes
0answers
17 views

Whether a given collection is a set or not [duplicate]

I originally knew that a set is a concept that has no definition. However, today, in abstract algebra class, the professor told us that the collection of all fields E such that E/F is an alegbraic ...
0
votes
0answers
35 views

Roots of unity in $\mathbb Q(\zeta_n)$

If $n$ is a positive integer then the roots of unity in $\mathbb Q(\zeta_n)$, with $\zeta_n=e^{\frac{2\pi i}{n}}$ is a cyclic group and is generated by $\zeta_{\tilde n}$, with ${\tilde ...
3
votes
1answer
95 views

Show that $K = \mathbb{Q}(\sqrt{p} \ | \ \text{p is prime} \}$ is an algebraic and infinite extension of Q

Show that $K = \mathbb{Q}(\sqrt{p} \ | \ \text{p is prime} \}$ is a algebraic and infinite extension on Q. Well, if i consider for every p prime, the polynomial p(x)=x^2−p, then p(x) is in Q(p√∣p is ...
1
vote
1answer
22 views

Question on separable field extenions

Hi I was given this question which I cannot express myself mathematically on so would indeed like the help and appreciate it I am given $ K/F $ is a finite field extension. I am required to show that ...
1
vote
1answer
18 views

Prove $\sigma_g(x) \in Aut(R(x)/R)$

Let $R$ be a field and let $R(x)$ be the field of rational functions in $x$ whose coefficients are in $R$. Let $g = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right) \in ...
0
votes
1answer
40 views

Let E be an extension of a field F of degree 2. Show there is an element with $\beta = x^2 - \alpha$

Let E be an extension of a field F of degree 2. Show there is an element with $\beta = x^2 - \alpha$. I wanna know if my approach is correct. If it's an extension of degree 2, then $F(\alpha)$ ...
0
votes
3answers
40 views

Factor this polynomial into linear factors with coefficients in $F = \mathbb{Q}(2^{1/3}, i\sqrt{3})$

The polynomial is this: $x^3 -2$ Okay, so first I can create my field extension. I can easily extend the field to $2^{1/3}$. And I know the elements of the extension of $\mathbb{Q}(2^{1/3})$ can be ...
2
votes
1answer
34 views

Transitivity of the discriminant of number fields

Let $M/L/K$ be a tower of number fields with discriminant of $M/K: d_M$ and of $L/K: d_L$. I would like to find a transitivity theorem for the discriminant and by letting $p_i$ and $q_i$ be integral ...
1
vote
1answer
59 views

Is $\mathbb{Q}(\sqrt{3})$ in someway related to Quotient ring?

I can't help but notice that they look exactly the same. For example: $\mathbb{Q}(\sqrt{3})$ = $\lbrace p + q\sqrt{3}:p,q \in \mathbb{Q}\rbrace$ That seems pretty much exactly an ideal. Only the ...
1
vote
2answers
23 views

Finding the conjugates, why can they argue this way?(exercise)

In one exercise I am supposed to find the conjugate of $\sqrt{2}+i$ over $\mathbb{Q}$. I found the answer by finding irr$( \sqrt{2}+i,\mathbb{Q})$, and then solving the polynomial finding all the ...
-2
votes
3answers
49 views

Notation Question(Abstract Algebra)

what does $\mathbb{Q}(\sqrt{3})$ mean?
4
votes
1answer
61 views

What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
2
votes
1answer
51 views

Does all splitting fields have characteristic 0?

Does all splitting field have characteristic 0? This may be a bad question, but I am wondering because the author of a book I am reading summarises a lot of properties when he is about to start with ...
-1
votes
0answers
22 views

Suppose that C is a companion matrix.why F is subfield F[C]?

Suppose that $C$ is a companion matrix of some irreducible monic polynomial $m\in F[x]$ ,$m(x)=m_0+m_1x+…+m_{n−1}x_{n−1}+x_n$. Consider the surjective homomorphism $F[x]→F[C]$ defined by $x↦C$. The ...
0
votes
0answers
35 views

Can it be proved that this extension is algebraic?

Assume that we have a field F, an extension field E of F, and both of them are contained in the algebraic clousure $\overline{F}$. Let E have the property that every automorphism of $\overline{F}$ ...
1
vote
0answers
17 views

Dimension of compositum of two fields, one of them Galois.

Let $L, K, F = L \cap K$ be fields such that $[L:F] , [K:F] < \infty$. $LK$ is defined as the compositum of the two fields(shortest field containing $K$ and $L$ in some algebraic closure). Also, ...
0
votes
1answer
37 views

Degree of $\mathbb{Q}(\omega)/\mathbb{Q}$ where $\omega^{3}=1$

I am working through Rotman's Galois Theory, and I came across an example that confused me a bit. Here is a screenshot of the problem: I am not sure, why the degree of ...
0
votes
1answer
23 views

Proof help: why is the constructed field a splitting field?

Here is my books definition of a splitting field: Note that it uses the word: smallest: In the last converse part of this theorem. I see that the field E created is a field that contains F(this is ...
7
votes
5answers
86 views

Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$?

Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$? I mean, $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ can be viewed as: $\mathbb{Q}[\sqrt{2}][\sqrt{3}]$, as polynomials of ...
1
vote
1answer
25 views

Is the compositum of $L_1$ and $L_2$ equal to $L_1[L_2]$?

In a course about Galois theory, there is the following definition : Let $L_1$ and $L_2$ be two subfields of the field $L$. We define the compositum $L_1L_2$ of $L_1$ and $L_2$ as the smallest ...
0
votes
0answers
47 views

Galois extension over power series fields

Let $K$ be a field, and $L$ be an algebraic extension of $K$. I think it is known that if $T$ is a finite extension of $K((X))$, then $T$ is complete with respect to the $X$-adic valuation, hence if ...
3
votes
1answer
30 views

Algebraic or not algebraic extension?

Suppose $F^{\prime}/F$ is an algebraic extension of fields, and that $F^{\prime}$ is a finite field extension of $K^{\prime}(x^{\prime})$, where $x^{\prime}$ is transcendental over $K^{\prime}$, and ...
4
votes
2answers
121 views

Is the intersection of finite index subfields finite?

Suppose that $K$ and $L$ are two fields contained in some larger field, and let $KL$ denote the smallest subfield of the ambient field containing both of them. If $KL$ is a finite extension of both ...
2
votes
1answer
70 views

Calculate the degree of the extension $[\mathbb{Q}(\cos(\frac{2\pi}{p})):\mathbb{Q}]$

Calculate the degree of the extension $[\mathbb{Q}(\cos(\frac{2\pi}{p})):\mathbb{Q}]$ where $p$ is a prime number. My thoughts are: I am lost My intuition says it has to be $ \frac{p-1}{2}$ and ...