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Questions tagged [extension-field]

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, normal/separable extensions... This tag often goes along with the abstract-algebra and/or field-theory tags.

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The Relationship Between Field Extension Tower and Tensor Product

Let $F$ be a field. Consider field extension $ F \subset L \subset K $. Let $ \alpha_1, \dots , \alpha_m $ be an $F$-basis of $ L $, $\beta_1, \dots, \beta_n$ an $L$-basis of $ K $. By the theory of ...
Long-Ping Li's user avatar
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Defining polynomial for a compositum of splitting fields

Let $L_1,...,L_n/K$ be finite separable field extensions. Then the compositum extension $L:=L_1\cdot ...\cdot L_n/K$ is also finite and separable. Thus by the primitive element theorem, there are $\...
Nicolas Banks's user avatar
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1 answer
85 views

Let $b=\sqrt[3] 2$, $c=-\frac{1}{2}+\frac{\sqrt {-3}}{2}$ with $a=bc$. Prove $b+a$ and $b-a$ has degree $3$ and $6$ over $\mathbb Q$, respectively.

Let $b=\sqrt[3] 2$, $c=-\frac{1}{2}+\frac{\sqrt {-3}}{2}$ with $a=bc$ in $\mathbb C$. Prove that $b+a$ and $b-a$ has degree $3$ and $6$ over $\mathbb Q$, respectively. My attempts: $b+a=b+bc=b(1+c)=\...
Fuat Ray's user avatar
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Inclusion of field extensions $\mathbb{Q}(\omega_p)$ and $\mathbb{Q}(\omega_{2p})$

Could you help me clear up a confusion? Either my following reasoning is wrong, or I should be able to show that $\omega_{2p} \in \mathbb{Q}(\omega_p)$, which does not seem correct. Degree of Field ...
lkksn's user avatar
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1 answer
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Natural way to extend the ring $\mathbb{Z} / p^k \mathbb{Z}$ so that the equation $x^2 + 1 \equiv 0 (\text{mod }p^k)$ has a solution

We know that for $p \equiv 3 (\text{mod }4)$, there is no solution to $x^2 + 1 \equiv 0 (\text{mod }p^k)$ for $k = 1, 2, \ldots$, by quadratic reciprocity. But can I embed the ring $\mathbb{Z} / p^k \...
John Jiang's user avatar
1 vote
1 answer
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Inertia field example of $ \mathbb{Q}_5(\sqrt[4]{50})$

Let $L = \mathbb{Q}_5(\sqrt[4]{50})$ and denote by $E$ the inertia field of the extension $L / \mathbb{Q}_5$. Write down a prime element $\pi_E $ of $ \mathcal{O}_E $ with $L = E(\sqrt{\pi_E})$. Can ...
Christian Schwacke's user avatar
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25 views

extension field in the subring $K[u]$

If $F$ is an extension field of a field $K$, $u, u_i \in F$, and $X \subset F$, then \begin{enumerate}[(i)] \item the subring $K[u]$ consists of all elements of the form $f(u)$, where $f$ is a ...
vivvv's user avatar
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On Frobenius–Schur indicator of real/complex representations

Let $G$ be a finite group with complex irreps $W_i$. Let $V$ be a real irrep of $G$. Denote $\chi_{W_i}$ and $\chi_{V}$ the corresponding characters. Each $V$ has three possibilities: Case 1: $\dim_{\...
khashayar's user avatar
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Is there a separable extension of degree 21?

Given the field $F= \mathbb{F}_3$ and a transcendental $t$, I am trying to find an intermediate field $$ F(t^{1/63}) \supset E \supset F(t) $$ where $[F(t^{1/63}): E]= 21$ and the extension is ...
Carlyle's user avatar
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1 answer
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Determining the Equality of Two Field Extensions

Let $F$ be a field of characteristic $0$. Let $F(\alpha)/F$ be a finite extension of degree not divisible by $3$. Is is true that $F(\alpha^3)=F(\alpha)$? If we assume that they are not equal, since $\...
Ty Perkins's user avatar
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1 answer
35 views

Easier way to show finite simple extensions have only finitely many intermediate fields

I was reading this post that concerns the idea that simple finite field extensions have only finitely many intermediate fields between them. I'm wondering why the following method hasn't been proposed:...
Grigor Hakobyan's user avatar
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2 answers
47 views

Purely inseparable field extensions - proving $\alpha^{p^m} \in F$ implies $m_\alpha = x^{p^m}-a^{p^m}$

I'm reading Isaacs' "Algebra: A Graduate Course" and I don't really understand the proof for the implication (2) $\Rightarrow$ (3) in Theorem 19.10 (page 298): Suppose $F\subseteq E $ is an ...
RatherAmusing's user avatar
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Are "other" algebraic extensions of $\mathbb{R}$ similar to $\mathbb{C}$? [duplicate]

If we define $\mathbb{R}[x] = \{ c_0 + c_1 x + \cdots + c_n x^n : n\in\mathbb{N}, \forall c_k \in \mathbb{R} \}$ as the set of polynomials with coefficients in $\mathbb{R}$ then I'm familiar with the ...
Patch's user avatar
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Finding the fixed field corresponding to Galois group $\textrm {Gal} (\mathbb Q (\root p \of 2, \zeta_p)$ for prime $p$.

I've already shown that this is the splitting field of the polynomial $x^p-2$ over $\mathbb Q$, and that the degree of extension of $L=\mathbb Q (\root p \of 2,\zeta_p)$ is $p(p-1)$. Let $H$ be the ...
RatherAmusing's user avatar
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Possibly inseparable extensions of $\mathbb{F}_p((t))$

I have a question on local fields. Some sources define a characteristic $p$ local field as of the form $k((t))$ where $k/\mathbb{F}_p$ is a finite extension. Some sources define it as a finite ...
Kai Wang's user avatar
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Determine if $\mathbb Q (\root4 \of 5i,\sqrt3)$ is a normal extension of $\mathbb Q$.

I've shown that $\mathbb Q (\root4 \of 5i,\sqrt3)$ is of degree 8, and that the roots of $x^4-5$ are $\root4 \of 5i,-\root4 \of 5i,\root4 \of 5,-\root4 \of 5$. I'm having trouble checking whether $\...
RatherAmusing's user avatar
1 vote
1 answer
129 views

Computing $[\mathbb{C}(x,y):\mathbb{C}(xy^n+x^ny,xy)]$ and similar extensions

The following three questions are similar to this question. For every natural numbers $k,l,m,n \geq 2$, define: $A_{n}=\mathbb{C}(xy^n+x^ny,xy)$. $B_{m,n}=\mathbb{C}(xy^m+x^my,xy^n+x^ny)$, in ...
user237522's user avatar
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1 vote
2 answers
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Concept confusions on simple extension and conjugates

I find an example of constructing the simple extension by choosing $p(x)=x^3-2$, so it gives $E_0=\frac{Q[x]}{\langle p(x)\rangle}$ as a field. As we know, there are 3 roots for $p(x)=0$ in $\mathbb{C}...
GGplay's user avatar
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Is $B$ flat as an $A$-module?

Suppose $A$ is an integral closed domain, and its quotient field is $K$. Suppose $L$ is a finite separable extension of $K$, and $B$ is the integral closure of $A$ in $L$. Is $B$ flat as an $A$-...
Born to be proud's user avatar
2 votes
1 answer
25 views

Preservation of linear independence under field extension

Let $L/K$ be a field extension and $x \in L$ be transcendental over $K$. Let $A, B$ be $K$-matrices of same size. I want to show that if $A$ has linearly independent columns over $K$, then $Ax+B$ has ...
Bubaya's user avatar
  • 2,254
1 vote
0 answers
47 views

Irreducibility of the $p^k$-th cyclotomic polynomial

I want to prove that the cyclotomic polynomial $\Phi_{p^k}$ is irreducible using Eisenstein (I know that every cyclotomic polynomial is irreducible, I am just trying this approach). I am exposing what ...
lkksn's user avatar
  • 131
3 votes
2 answers
349 views

A question about Hilbert Theorem 90 and Artin-Schreier Theorem

I'm reading Lang's "Algebra" and there's a passage in the proof of Theorem 6.3 pg.290 (namely Hilbert's Theorem 90 additive form) for which I can't find a justification, if anyone could ...
F. Salviati's user avatar
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0 answers
72 views

$\overline{\mathbb{F}_2}$ does not contain a primitive 10th root of unity

I need to prove/disprove the following statement: Every algebraically closed field $K$ contains a 10th root of unity. I don't think the statement is true. My counterexample is as follows: Let's take ...
muhammed gunes's user avatar
4 votes
2 answers
118 views

Alternative method : $\mathbb{Q}(\zeta_3+\sqrt[3]{7})=\mathbb{Q}(\zeta_3,\sqrt[3]{7})$

Let $\zeta=\zeta_3$ be a third root of unity. I want to proof that $\mathbb{Q}(\zeta+\sqrt[3]{7})=\mathbb{Q}(\zeta,\sqrt[3]{7})$. One inclusion is clear that is : $\mathbb{Q}(\zeta+\sqrt[3]{7})\subset\...
muhammed gunes's user avatar
1 vote
0 answers
63 views

What is the Galois Group of $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$?

I am studying for some algebra qualifying exams over the summer and I am stumped on the title question: What's the Galois Group for $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$? Here's what I've got so far: ...
S.H.'s user avatar
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1 vote
1 answer
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Problem (9.19) - Character theory of finite groups (Martin Isaacs)

I am currently reading chapters $9$, $10$ et $15$ of Isaacs' character theory of finite groups and I am stuck with this question on splitting field. In this case, I have to found a splitting field for ...
GC.'s user avatar
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0 answers
70 views

How to find the Galois group of an infinite Galois extension? For example $\mathbb{Q}(\mathbb{\sqrt{\mathbb{Q}}})$. [duplicate]

I know that this is a very broad question, but I am especially interest in the case of the Galois extension $\mathbb{Q}(\sqrt{\mathbb{Q}}) = \mathbb{Q}( \{ \sqrt{-1} \} \cup \sqrt{p} \mid p \text{ ...
Cosima's user avatar
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2 votes
1 answer
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An exercise about abelian Kummer extensions

I'm trying to do this problem about abelian Kummer extensions: Image transcript and my attempts are below: Let $K/F$ be a Galois extension with Galois group $G=\operatorname{Gal}(K/F)$ of order $n$. ...
hbghlyj's user avatar
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2 votes
1 answer
68 views

If $a \in \mathbb{C}$ is algebraic over $\mathbb{Q}$, determine the minimal polyn of $\sqrt{a}$ over $\mathbb{Q}$ when $\sqrt{a} \notin \mathbb{Q}(a)$

This exercise is broken down into parts: I first proved that $\sqrt{a}$ is algebraic over $\mathbb{Q}$. Given that $a \in \mathbb{C}$ is algebraic over $\mathbb{Q}$, let $m_a(x)$ be the minimal ...
Camilo Diaz's user avatar
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1 answer
78 views

How to prove that $[\mathbb{Q}(2^{1/3},3^{1/4}):\mathbb{Q}]=12$

I know that $$[\mathbb{Q}(2^{1/3},3^{1/4}):\mathbb{Q}]=[\mathbb{Q}(2^{1/3},3^{1/4}):\mathbb{Q}(3^{1/4})]\cdot[\mathbb{Q}(3^{1/4}):\mathbb{Q}],$$ and $$[\mathbb{Q}(2^{1/3},3^{1/4}):\mathbb{Q}]=[\mathbb{...
Camilo Diaz's user avatar
3 votes
1 answer
58 views

What are good sources for examples of field extensions?

I need to understand field extensions and Galois theory. I have read about the theory. But I think to digest the subject, i need to see lots of (non)normal, (in)separable, (in)finite dimensional, (non)...
boyler's user avatar
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0 answers
38 views

Primitive elements of the rationals function extension [duplicate]

I want to prove this: Let $k$ a field, $E:=k(x)$ the quotient field of $k[x]$ and $u\in E$. Show tha $E=k(u)$ if, and only if, $u=\frac{ax+d}{cx+d}$ for some $a,b,c,d\in k$ such that $ad-bc\neq 0$. I ...
Diego Ramírez's user avatar
2 votes
1 answer
128 views

Irreducibility of a Polynomial with Prime Exponents

Let $f(x) = (x^p - a_1)(x^p - a_2) \ldots (x^p - a_{2n}) - 1$ where $a_i \geq 1$ are distinct positive integers where at least two of them are even, and $n \geq 1$ is a positive integer and $p$ is ...
math.enthusiast9's user avatar
1 vote
0 answers
33 views

Degree of sum or product of linearly independent algebraic numbers.

Let $\alpha,\beta \in \mathbb{A}$ such that $\alpha,\beta$ are linearly independent over $\mathbb{Q}$ and $deg(\alpha)=k_{1}, deg(\beta)=k_{2}$. Can I deduce that $deg(\alpha \beta)=lcm(deg(\alpha),...
Math Admiral's user avatar
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0 answers
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Proof of the existence of algebraic closure

I am taking an introductory course on Galois theory, and we recently covered the following theorem: For every field $\mathbb{F}$, there exists a field extension $\mathbb{L}$ which is algebraically ...
A.P's user avatar
  • 11
2 votes
2 answers
132 views

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

How can I show that the extension $\mathbb{Q}(\sqrt[3]{5} \cdot \sqrt{2})$ is contained in the extension $\mathbb{Q}(\sqrt[3]{5} + \sqrt{2})$? I need to show that $\sqrt[3]{5} \cdot \sqrt{2}$ can be ...
lkksn's user avatar
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1 vote
0 answers
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Let $E$ be an extension of $F$ and let $A = \{e \in E: e$ is algebraic over $F\}$. Show that $A$ is a subfield of $E$ containing $F$.

Let $E$ be an extension of $F$ and let $A = \{e \in E : e$ is algebraic over $F \}$. Show that $A$ is a subfield of $E$ containing $F$. First, clearly $A \subseteq E$. Also since $F$ is algebraic over ...
Grigor Hakobyan's user avatar
3 votes
1 answer
96 views

Prove that $[\mathbb{Q}(\sqrt{3}, \sqrt[3]{3}, \sqrt[5]{3}, \xi_3,\xi_5) \colon \mathbb{Q}] = 240$.

I have already proven that $[\mathbb{Q}(\sqrt{3}, \sqrt[3]{3}, \sqrt[5]{3}) \colon \mathbb{Q}] = 30$ and $\sqrt{3} \not\in \mathbb{Q}(\xi_5)$. Consequently, $[\mathbb{Q}(\xi_5,\sqrt{3}) \colon \...
David's user avatar
  • 165
3 votes
1 answer
126 views

Is the imaginary unit $i$ contained in $\mathbb{Q}(e+i)$?

Is the imaginary unit $i$ contained in $\mathbb{Q}(e+i)$? $i \in \mathbb{Q}(e+i)$? Intuition tells me $i$ is not contained in $\mathbb{Q}(e+i)$ because it should somehow contradict the fact that $e$ ...
Rainbow's user avatar
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0 votes
3 answers
87 views

How to prove $\sqrt{2} \notin \mathbb{Q}(\sqrt[3]{2})$ using elementary techniques?

Prove $\sqrt{2} \notin \mathbb{Q}(\sqrt[3]{2})$. My try by contradiction: Assume there exist $a,b,c \in \mathbb{Q} $ such that: $\sqrt{2}=a + b \sqrt[3]{2} +c \sqrt[3]{4}$. Squaring both sides, we ...
Math Admiral's user avatar
  • 1,416
2 votes
1 answer
69 views

Subfields of splitting field of $x^4+25$ over $ℚ$.

Let $F$ be the splitting field of the polynomial $x^4+25$ over $ℚ$. List all subfields in $F$ and the corresponding subgroups in the Galois group. is problem $1$ on this pdf. The solution is: As we ...
hbghlyj's user avatar
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0 answers
38 views

If $\alpha$ is algebraic over $F$, then it is algebraic over any extension field $L$ of $F$.

The photo attached is from the book Abstract Algebra by Dummit&Foote. I didn’t understand it at all. How do we know that $L$ contains the element $\alpha$. In other words, what does $\alpha $ have ...
boyler's user avatar
  • 375
3 votes
1 answer
105 views

non-isomorphic number fields

Show that following 2 polynomial do not generate isomorphic number number fields: $P_1(x)= x^3+10x+1$ $P_2(x) = x^3-8x+15$ I see $Disc(P_1)=Disc(P_2)=-4027$ and both of them have a real root and 2 ...
mshj's user avatar
  • 520
2 votes
1 answer
95 views

If $F(\alpha) = F(\alpha^2)$, then $\alpha$ is algebraic over $F$

Let $K$ be an extension of a field $F$ and let $\alpha \in K$. If $F(\alpha) = F(\alpha^2)$, then $\alpha$ is algebraic over $F$. I have an idea about how to do this that doesn't trivialize the ...
Grigor Hakobyan's user avatar
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0 answers
27 views

Proof of equivalent definitions of normal extensions

Let $F \supset k$ be an algebraic extension. We have these two characterizations of a normal extension: a) For every $k$-homomorphism $\sigma: F \to \overline{k}$, we have $\sigma(F) \subset F$ b) If ...
lkksn's user avatar
  • 131
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0 answers
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Sum of nth power of some of the roots of irreducible polynomial over $ \mathbb{Q}$ is in $ \mathbb{Q}$

So i know that for a splitting field K over $ \mathbb{Q}$ of the polynomial f(x), where a,b,c,d are the roots of f(x). Taking the following sum $ a^n +b^n+c^n +d^n $ is in the FixGal(K,$ \mathbb{Q}$) ....
NoetherBoy 's user avatar
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0 answers
69 views

Reducibility of $x^2-7$ over $\mathbb{Q}(\sqrt[5]{3})$

Suppose for a contradiction that $x^2-7$ is reducible over $\mathbb{Q}(\sqrt[5]{3})$. Then $\sqrt{7}\in\mathbb{Q}(\sqrt[5]{3})$. It follows that $\mathbb{Q}\subset\mathbb{Q}(\sqrt{7})\subset\mathbb{Q}(...
spinosarus123's user avatar
0 votes
1 answer
70 views

Check when a positive square Root $\sqrt{d}$ is contained in Cyclotomic Field

Let $\zeta_n $ be a primitive root of unity generating the cyclotomic field $\Bbb Q(\zeta_n)$. Is/are there quick and/or "standard" techniques" to check if a given real quadratic roots $...
user267839's user avatar
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2 votes
1 answer
38 views

Let $F \supset K \supset L$ be fields with orders less than 100 and do not include an element $x\neq 1$ that $x^5=1$. Find the order of $F$.

Let $F \supset K \supset L$ be fields with order less than 100 and do not include an element $x\neq 1$ that $x^5=1$. Find the order of $F$. my attempt I used the Tower Law where we have $ [F:L]=[F:K][...
White Give's user avatar
0 votes
1 answer
48 views

Why is the subfield of $\mathbb{Q}(\zeta_p)$ of index $2$ expressible in terms of the sum of $\zeta_p$ to the power of all quadratic residues mod $p$?

Let $p$ be an odd prime, $\zeta_p = e^{2 \pi i / p}$. I've been playing around with calculating the intermediate fields of $\mathbb{Q}(\zeta_p)$. I know that $\mathrm{Gal}(\mathbb{Q}(\zeta_p) / \...
Robin's user avatar
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