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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1answer
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If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?

Let's consider a field $F$ and $\alpha,\beta\in\overline{F}-F$ (where $\overline{F}$ is an algebraic closure). If $F(\alpha)=F(\beta)$, is it true that $\alpha$, $\beta$ have the same minimal ...
3
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3answers
59 views

Irreducible polynomial over $\mathbb{Q}(\zeta)$

Show that the polynomial $f(x)=x^5-2$ is irreducible over $\mathbb{Q}(\zeta)$, where $\zeta=e^{2\pi i/5}$. I tried show that the roots of polynomial $f(x)=x^5-2$, $$\sqrt[5]{2}, \zeta\sqrt[5]{2}, ...
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1answer
19 views

I have to show it is isomorphic to $K = GF(p^{kd})$ [closed]

Suppose $F = GF(p^k)$ is a finite field. I know $F[C]$ is a field extension of $F$ with degree $d = \deg m$, and I have to show it is isomorphic to $K = GF(p^{kd})$ (where $C$ is a companion matrix ...
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0answers
28 views

Intermediate “prime” extensions [Rotman]

Problem Assume $F$ contains the $k$th roots of unity, and let $R=F(\alpha)$, where $\alpha$ is a root of $x^k-a$ for some $a\in F$. Prove that there exist intermediate fields $$F=K_0\subset ...
1
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1answer
23 views

Isomorphism of quadratic extensions (of a number field)

I think we agree that two (squarefree) quadratic extensions of $\mathbb Q$, say $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 3)$ are not isomorphic. Now consider the following tower of fields ...
6
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1answer
73 views

Integral basis of $\mathbb{Q}(\theta)$, where $\theta^3-\theta-4=0$

I am working on the text book "algebraic number theory" by Jurgen Neukirch(P15, exercise 6). To prove the integer basis is$ \{1, \theta, \frac{\theta^2+\theta}{2}\}$. After a long and tedious ...
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votes
2answers
42 views

Degree of minimal polynomial

The minimal polynomial of $a$ over $\mathbb{Q}$ is quadratic. The minimal polynomial of $b$ over $\mathbb{Q}$ is cubic. Is the minimal polynomial of $a+b$ necessarily of degree $6$? If so, what is ...
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4answers
101 views

Show that $\mathbb{Q}(\sqrt{5}+\sqrt[3]{2})=\mathbb{Q}(\sqrt{5},\sqrt[3]{2})$

I've got that $[\mathbb{Q}(\sqrt{5}+\sqrt[3]{2}):\mathbb{Q}] \in \{1,2,3,6\}$ because it's going to divide $[\mathbb{Q}(\sqrt{5},\sqrt[3]{2}):\mathbb{Q}]=6$. Clearly it is not $1$. I want to show that ...
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1answer
41 views

Why is is $K(\alpha,\beta)/K(\alpha)$ algebraic if $K(\alpha,\beta)/K(\beta)$ is algebraic? [duplicate]

Let $K$ be a field, and let $\alpha$ be transcendental over $K$ and algebraic over $K(\beta)$. We have a Hasse diagram of field extensions Now, by reduction to absurdity $\beta$ must be ...
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0answers
25 views

Exercise about cyclic field extension

I am having hard time to solve following exercise. Let $\Omega$ be the algebraic closure of a field $k$. a) Suppose that every finite extension of $k$ is cyclic. Prove that it exists $\sigma \in ...
2
votes
1answer
35 views

Number of Galois conjugates

Let $L/\mathbb Q$ be a (finite) Galois extension of degree $n$ with Galois group $\Gamma$. We know that there is a primitive element or generator $\alpha$ of this extension. My question: Is the ...
1
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1answer
45 views

How does one construct the Galois field extension $GF((2^2)^3)$?

Looking at past exam question, one asks us to construct a Galois field extension $GF((2^2)^3)$ whenever the primitive irreducible polynomial $p(X) = X^3 + \alpha X^2 + \alpha X + \alpha \in ...
4
votes
0answers
54 views

How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$?

$\mathbb{F}/\mathbb{Q}$ is a finite extension. Denote $\dim_{\mathbb{Q}}\mathbb{F}=n$. How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$? Thoughts: if ...
4
votes
2answers
57 views

Why is this Galois group abelian?

Consider the field extension $\mathbb Q(\zeta_3,\sqrt[3]2,\zeta_8)/\mathbb Q(\zeta_3)$, with intermediate fields $\mathbb Q(\zeta_3,\sqrt[3]2)$ and $\mathbb Q(\zeta_3,\zeta_8)$. Denote ...
2
votes
1answer
32 views

Cyclic extension without primitive root of unity

Let $F$ be a field that doesn't contain a primitive fourth root of unity. Let $L = F(\sqrt a)$ for some $a \in F - F^2$ and let $K=L(\sqrt b)$ for some $b \in L - L^2$. If we have $N_{L/F}(b) ...
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1answer
12 views

Splitting of primes terminology doubt

What do we mean when we say that a given prime $p$ splits completely in an algebraic extension of $\mathbb Q$? Are we talking about the splitting of prime ideals into unique factors? And, in that ...
2
votes
1answer
29 views

Field extension of prime degree

Question: Let $L$ be the extension of the field $K$ such that $[L:K]=p$, where $p$ is a prime number, and $\alpha \in L$. Prove that $K(\alpha)=K$ or $K(\alpha)=L.$ Proof: From $$ \alpha \in L ...
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0answers
32 views

Find and show primitive element of field extension

I have the polynomial $\ f(x)= x^{15}-3 $ and want to find the splitting field. The splitting field is surely $\Bbb Q$ adjoined the roots of the polynomial. I want to write the splitting field as ...
0
votes
0answers
34 views

What is the possible number (supremum) of subfields of $\mathbb{F}$?

Let $\mathbb{F}$ be field. it is a finite dimensional extension over $\mathbb{Q}$. So let $B=\{v_1, v_2, v_3, ... , v_n\}$ be a basis for $\mathbb{F}$ over $\mathbb{Q}$. From the finite dimension ...
2
votes
5answers
70 views

looking for the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$ with $\alpha=\sqrt[3]{2}$

I have $\alpha=\sqrt[3]{2}$ and want to calculate the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$. There's a hint which tells me to look at the minimal polynomial $m_\alpha$ of $\alpha$ over $\Bbb ...
4
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1answer
48 views

Showing that $\mathbb{Q}(\sqrt{2},\sqrt{3}, \sqrt{(9 - 5\sqrt{3})(2-\sqrt{2})})$ is normal over $\mathbb{Q}$, and finding its Galois group

If $K=\mathbb{Q}(\sqrt{2},\sqrt{3}, u)$, where $u^2 = (9 - 5\sqrt{3})(2-\sqrt{2})$, show that $K/\mathbb{Q}$ is normal, and find $\operatorname{Gal}(K/\mathbb{Q})$. I found that the minimal ...
0
votes
1answer
41 views

finding intermediate fields of field extensions: simple method vs. Galois theory

I am looking for a straightforward way to find all intermediate fields of a field extension. Let's take the splitting field of $X^3-2$ over $\Bbb Q$ as an example. If we adjoin the roots of $X^3-2$ ...
3
votes
1answer
44 views

Galois group of primitive root of unity/ field extensions

Suppose $\alpha = \omega + \omega^7 + \omega^{11}$, where $\omega$ is a primitive root of unity of order 19. I want to determine the Galois group of $\mathbb{Q}(\alpha) / \mathbb{Q}$. Because ...
0
votes
4answers
35 views

Basis of $\mathbb{Q}[\sqrt[3]{2}]$

How do I prove that $1, \sqrt[3]{2}, (\sqrt[3]{2})^2$ is a basis of $\mathbb{Q}[\sqrt[3]{2}] = \{ a + b \sqrt[3]{2} + c (\sqrt[3]{2})^2\: a,b,c \in \mathbb{Q} \}$. It's one of these cases where the ...
4
votes
1answer
49 views

Is the primitve element of $\mathbb{Q}[\alpha_1, \alpha_2, \ldots]/\mathbb{Q}$ always $\alpha_1 + \alpha_2 + \cdots$?

I have dealt with a number of algebraic field extensions $\mathbb{Q}[\alpha_1, \alpha_2, \ldots]/\mathbb{Q}$ and the primitive element was always $\alpha_1 + \alpha_2 + \cdots$. Is this generally true ...
1
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0answers
76 views

Extensions of a field?

Prove that there are infinitely many degree 5 extensions of $\mathbb{F}_{121}(x)$. I know that $\mathbb{F}_{121}$ is isomorphic to the splitting field of $x^{121}-x$ over $\mathbb{F}_{11}$, but I'm ...
4
votes
3answers
82 views

What elements may I adjoin to $\mathbb{Q}[\sqrt{3}]$ in order to get to $\mathbb{Q}[\sqrt{7+\sqrt{3}}]$

The field extension $\mathbb{Q}[\sqrt{7+\sqrt{3}}]/\mathbb{Q}$ has degree four and $\sqrt{7+\sqrt{3}}$ is a primitive element. I'm interested in dividing this into two successive field extensions of ...
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0answers
82 views

Is $\mathbb{Q}(\sqrt{2})\subset\mathbb{Q}\left(\sqrt{2},i\sqrt[3]{2}\right)$ a Galois extension?

Consider the field extension $$\mathbb{Q}(\sqrt{2})\subset\mathbb{Q}\left(\sqrt{2},i\sqrt[3]{2}\right)$$ Is it Galois? I can't quite find the order of the automorphism group.
0
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1answer
32 views

Irreducible polynomial of $\sqrt{2}+\sqrt{7}$ on $\Bbb{Q}$.

I would like to find the irreducible polynomial on $\Bbb{Q}$ of $\sqrt{2}+\sqrt{7}$. How can I do that ? First time I see this kind of question, I can find a polynomial $X^2-2$ witch $\sqrt{2}$ is a ...
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0answers
13 views

Bilinear forms on vector spaces over field extension

Could anyone please help? Let K be an algebraic number field, and consider the symplectic form on $B: K^2 \times K^2 \to \mathbb{Q}$ given by $B((k_1,k_2),(l_1,l_2)) = Tr_{K/ \mathbb{Q}}(k_1 l_2 - ...
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1answer
15 views

Question on the normal closure of a field extension

Hi all I was given the following question in my field theory class which I am stuck on: I am given $ K/F $ a finite extension of fields. I am asked to show the existence of an $ K \subset L $ such ...
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3answers
42 views

Field extensions for polynomial $T^3+2T +1$ in $\mathbb F_3$

I have the following polynomial over $\mathbb F_3$: $$ f(T) = T^3+2T+1 $$ I would like to find out a field extension in order to add the roots of this polynomial. Edit: by defining $\alpha\not\in ...
0
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1answer
73 views

How can I compute the discriminant of $\mathbb{Q}(\sqrt{2}+\sqrt{5})$?

How can I compute the discriminant of $\mathbb{Q}(\sqrt{2}+\sqrt{5})$? I get stuck in this exercise of chapter 12 of textbook "A classical introduction to modern number theory" very long time... ...
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1answer
45 views

Finding a primitive fifth root of unity modulo $81$ using a specific method.

I want to find a fifth root of unity modulo $81$ using a suggested method from the book (I can't come up with any other good method anyway). It is given that $x^4+x+2 \in \mathbb{F}_3[x]$ is ...
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2answers
165 views

Galois extension of degree 75

I'm studying Galois Theory and I stumbled upon this question: Let $E/K$ be a Galois extension of degree $75$. Show that there exists an intermediate subfield $K \subsetneq F \subsetneq E$ such ...
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2answers
49 views

Show that $f=x^3+7x+5$ has no roots in $\mathbb {Q}(\sqrt[4]{2})$

Show that $f=x^3+7x+5$ has no roots in $\mathbb {Q}(\sqrt[4]{2})$. I'm given a hint: suppose $\alpha$ is a root of $f=x^3+7x+5$ and $\alpha\in\mathbb{Q}(\sqrt[4]{2})$, compute ...
4
votes
4answers
111 views

$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$

Prove that $$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$$ I proved for two elements, ex, $\mathbb{Q}\left ( \sqrt{2},\sqrt{3}\right ...
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1answer
33 views

Linear combination of matrices over finite and infinite fields

Let $F\subset K$ be the fields. Let $A_1,\ldots, A_m$ be the $n\times n$ matrices over the field $F$, and $c_1,\ldots,c_m\in K$ such that $c_1A_1+\cdots+c_mA_m$ is invertible. How to prove that for ...
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1answer
72 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 $P,Q \in \mathbb{F}_p[t]$. ...
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1answer
32 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 ...
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1answer
31 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
80 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 ...
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3answers
83 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
36 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
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1answer
44 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
67 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
36 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
21 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 ...