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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4
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2answers
87 views

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

Showing that $\mathbb{Q}(\sqrt{2},\sqrt[3]{3})$ has degree 6 over $\mathbb{Q}$ is straighforward: It contains $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt[3]{3})$ which are degree 2 and 3 over $\...
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1answer
14 views

When does conjugation in simple field extensions coincide with the identity?

Suppose I have a field $F$ and a simple extension of $F$, $F(\alpha)$, and let's assume that $\alpha^2 \in F$. Now take the "conjugation" automorphism $\sigma:F(\alpha) \to F(\alpha)$, $\...
2
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1answer
41 views

Trouble understanding the general polynomial of degree $n$

Herstein defines the general polynomial of degree $n$ over a field $F$ as the polynomial $p(x) = x^n + a_1x^{n-1} + \cdots + a_n \in F(a_1, \cdots, a_n)[x]$. From here, since it was previously shown ...
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3answers
22 views

Given f(x) in k[x], is it possible to find two field extensions K/k and K'/k such that f has two different factorizations as linear polynomials?

By Kronecker's theorem, given a field $k$ and $f(x)$ a polynomial with coefficients in $k$, there exists a field $K$ containing $k$ as a subfield and with $f(x)$ a product of linear polynomials in $K[...
2
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1answer
50 views

Can the minimal polynomial have multiple roots?

Let $F$ be an extension field of $K$. I'm trying to think of an example of an element $\alpha$ which is algebraic over $K$ and whose minimal polynomial over $K$ has a root of multiplicity $> 1$ in ...
0
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1answer
20 views

How to prove $\big[\text{Frac}\left(\mathbb{F_p}[x]\right):\mathbb{F} \big]=\infty$ for a prime number $p$?

Let $p$ be a prime number and consider the ring $\mathbb{F}_P=\{0,1,\ldots,p-1\}$ of integers modulo $p$, which is a field. It follows that the polynomal ring $\mathbb{F}_p[x]$ is an integral domain, ...
2
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3answers
55 views

Field extension of $\Bbb Q$ of degree $4$ has reals

Suppose $L / \mathbb{Q}$ is a field extension with $[L : \mathbb{Q}]$ = 4, with $L \not\subset \mathbb{R}$. Is it true that $L \cap \mathbb{R} \neq \mathbb{Q}$? If not, is it true if $L / \mathbb{Q}$ ...
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1answer
62 views

What are the elements of $\mathbb{Q}(\sqrt[3]{2}+\sqrt{3})$?

I would like to calculate the elements of $\mathbb{Q}(\sqrt[3]{2}+\sqrt{3})$. I know that the elements of $\mathbb{Q}(\sqrt[3]{2})$ have the form of ${a+b\sqrt[3]{2}+c\sqrt[3]{4}}$, where a,b,c $\in \...
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0answers
42 views

What is the intuition behind the definiton of a seperable field extension?

so I am currently studying separable extensions in John Fraleigh's abstract algebra book and have come upon this definiton of a separable field extension: A finite extension $E$ of $F$ is a separable ...
3
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1answer
51 views

A property of fields of characteristic $p$ and their extensions

I am currently stuck on the following problem, taken from Herstein's Topics in Algebra, 2nd ed.. It reads: If $F$ is of characteristic $p \neq 0$ and if $K$ is a finite extension of $F$, prove that ...
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0answers
29 views

Is that $[\mathbb{C} : \mathbb{R}] = 2$ and is that $[\mathbb{C} : \mathbb{Q}]$ infinite?

Since, $\mathbb{C} = \mathbb{R}(i)$ where $i = \sqrt{-1}$. So, $[\mathbb{C} : \mathbb{R}] = [\mathbb{R}(i) : \mathbb{R}] = 2$. Which $\mathbb{C}$ is a finite field extention of $\mathbb{R}$. Since, ...
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0answers
21 views

If $\beta$ is transcendental over $F$, then is that $F(\beta)$ must be an infinite extention of $F$?

Let $F$ be fields. If $\beta$ is transcendental over $F$, then is that $F(\beta)$ an infinite extention of $F$ ? Or $F(\beta)$ can be an finite extention of $F$ ?
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0answers
14 views

Prove that $[M_{j+1} : M_j] \le 2$ given $[L_{j+1} : L_j] = 2$ and $M_j = L_j \cap \mathbb{Q}(\alpha).$

I am reading a textbook and I believe it makes a claim that is either false or not fully justified. By the definition of $Q_{py}$ there is a tower $$\mathbb{Q} = L_0 ⊆ L_1 ⊆ \dots ⊆ L_n ⊇ \mathbb{Q}(\...
1
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1answer
37 views

Algebraic Closure is Unique up to Isomorphism in Dummit Foote

I know that there are many proofs of this, but I am wondering if someone could help me understand what is "swept under the rug" in Dummit and Foote. I couldn't find anything about that on ...
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0answers
22 views

Intermediate fields $K((K^{1/p^\infty} \cap L)^{p^\infty})$ and $K(K(L^{p^\infty}))^{1/p^\infty} \cap L $ of field extension $L/K$

Let $K$ be a field of characteristic $p >0$ and $L/K$ an algebraic extension. We can assoicate to these fields two canonical operations: The perfect closure a $K^{1/p^{\infty}} := \bigcup_{i \le 1}...
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0answers
13 views

proving a finite field extension separable

$F/K$ be a finite field extension and $F=K(F^p)$. Then show that $F/K$ is separable extension. I have been asked to solve the above problem. The problem seems to miss some information. It must be ...
3
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4answers
181 views

Four dimensional field over complex numbers

A guy in Facebook claims he's come up with an algebraic field extension to the complex plane. He's defined the unit multiplications as $i^2=-1$, $j^2=i$ and $k^2=-i$. This implies that $ij=ji=k$, $ik=...
3
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2answers
39 views

Degree of Extension from $\mathbb{Q}$ to $\mathbb{Q}(i, \sqrt[4]2)$

I wanted to find the degree of extension from $\mathbb{Q}$ to $\mathbb{Q}(i, \sqrt[4]2)$. The minimal polynomial of $\sqrt[4]2$ over $\mathbb{Q}$ is $f(x)=x^4-2$. Now the roots of $f(x)$ are $\{\sqrt[...
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2answers
53 views

Irreducible polynomial of algebraic element [closed]

I am out of ideas. Any hint of how can I compute the irreducible polynomial of $\sqrt[3]{4}+\sqrt[3]{2}-1$ over $\mathbb{Q}$? I only get a lot of computations to nowhere.
4
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1answer
32 views

Comparison of degrees of field extensions

Let $M$ be a field, and let $F$ and $K \subset L$ be subfields of $M$. Do we have the inequality $$[L \cap F \colon K \cap F] \leq [L \colon K] \, \text{?}$$ Using the multiplicativity of degrees, it ...
3
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1answer
36 views

Existence of Intermediate field $F$ of $E$ and $G$ such that the $[G: F] < \infty$

if $G$ is an infinite extension of $E$ $$E \subset G $$ Then does there exists an extension $F$ of $E$ such that $G$ is finite extension of $F$ $$E \subset F \subset G$$ I think we need to remove some ...
3
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1answer
60 views

Transcendence basis as subset of generators

Let $K \subset L = K(a_1,...,a_n)$ be a field extension finitely generated as $K$-algebra with transcendence degree $\operatorname{Trdeg}_K(L):= m \le n$. It is well known that the choice of a ...
0
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1answer
44 views

Decomposition of Field extensions and Transcendence basis

Let $K \subset L$ be arbitrary extension of fields and assume (for the first time) it has a finite transcendence degree $n:= \operatorname{Trdeg}_K(L)$. Let $\{a_1,...,a_n\}$ it's transcendence basis. ...
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2answers
51 views

Does it matter whether we consider field extensions as true supersets of a field?

I am trying to wrap my head around field extensions, and am using Pinter's abstract algebra. He presents a basic theorem on field extensions, namely that given any field $F$, and any polynomial $a(x)$ ...
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0answers
28 views

Understanding field automorphisms

I am studying splitting fields, but have realized that I have a conceptual gap when it comes to field automorphisms. More specifically, if $F$ is a field and $\overline{F}$ is its algebraic closure so ...
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0answers
54 views

A solvable galois group implies radical extension

THEOREM 18.18 Let K be a field of characteristic 0 and let L : K be a finite normal extension with soluble Galois group G. Then there exists an extension R of L such that R : K is radical. I'm ...
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1answer
29 views

Field automorphisms send roots into roots

In the post Prove that a field automorphism sends a root into a root, the following problem is discussed: If $E$ is an extension of $F$, $f(x) \in F[x]$ and $\phi$ is an automorphism of $E$ leaving ...
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0answers
30 views

Homomorphism between the Field F and the Quotient Ring of Polynomial Ring

I'd like to show for a field $F$ and ideal $I$, the mapping $\phi: F \rightarrow F[x]/I$ defined by $\phi(a)=a+I$ is a ring homomorphism. Showing $\phi(a+b)=\phi(a)+\phi(b)$ is rather straightforward. ...
4
votes
1answer
57 views

Adjoining element in a finite field $\mathbb F_{p^n}$ to $\mathbb F_p$ to get $\mathbb F_{p^6}$

Given that $p$ is a prime and $n$ is a natural number, I want to know how many $\alpha$ are in $\mathbb F_{p^n}$ such that $\mathbb F_p (\alpha) = \mathbb F_{p^6}$. I know that $[\mathbb F_{p^6} : \...
1
vote
1answer
34 views

Is $G(\mathbb{Q}(\sqrt[3]{2}), i\sqrt{3})/\mathbb{Q}(i\sqrt{3}))\simeq \left \langle \mathbb{Z_3}, + \right \rangle$?

I wonder I have some misunderstanding of a concept of automorphism that leaving some field fixed. \ The Problem in the text(Fraleigh, p.402, 7th) is: Referring to Example 50.9, show that $$G(\mathbb{...
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0answers
47 views

Example of a purely inseparable extension

Let $L/K$ be an algebraic extension of fields. We say $L/K$ is a purely inseparable extension if there is only one embedding of $L$ into an algebraic closure of $K$. Suppose this is the case. Then ...
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0answers
22 views

Issue in the proof showing that the splitting fields are normal.

I am using the following definition of a normal extension: $K \subset L$ is normal if for all $\Omega$ with $K \subset \Omega$, for all $K$ embeddings $x_1:L\rightarrow \Omega, x_2:L\rightarrow \Omega$...
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2answers
55 views

Factorize $f(X)=X^4+1$ on $L$.

Let $L= \mathbb{Q}[Y]/(Y^4+1)$. Factorize $f(X)=X^4+1$ on $L$. My attempt: We know that $\alpha = \overline{X}=X+(X^4+1)$ is one of the root of $f(X)$ in $L$. Now using the relation $\alpha^4+1=0$ and ...
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0answers
28 views

Inverse image of a “rational map” is “rational”

Let $K'\subset K$ be fields, $V$ be a right $K$-vector space and $V'$ a $K'$-subspace of $V$ such that the $K$-linear mapping $\lambda:V'\otimes_{K'}K\rightarrow V$ such that $\lambda(x'\otimes \xi)=x'...
0
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1answer
25 views

Extensions with the same degree are $K-$isomorphic?

Suppose we have two simple algebraic extensions $K(a)$, $K(b)$ over a field $K$ (I am interested in case $\mathrm{char}(K)=0$ but I guess there's no difference) If they have the same (finite) degree ...
1
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1answer
37 views

$F(a,b)$ separable if $F(a)$ and $F(b)$ are separable

Let $F$ be a field and $F(a)$, $F(b)$ simple algebraic separable extensions of $F$. Is then $F(a,b)$ separable over $F$? I was trying to use the transitivity of finite separable extensions (since $F(a)...
2
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0answers
34 views

If two elements generate the same extension, is there a relationship between their minimal polynomials?

Fix some base field $F$ and some $\alpha,\beta$ such that $F(\alpha) = F(\beta)$. We know that there exists some minimal polynomial $p(x)$ such that $\alpha$ satisfies $p(x)$ and any polynomial in $F[...
0
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1answer
27 views

Basis for field extension $\mathbb{Q}(\sqrt{d})$ with composite d

Every time I see an example of a basis for $\mathbb{Q}(\sqrt{d})$ in my book, $d$ is prime. Is there a reason for this? Let $d=p_1\times p_2$ where $p_1$ and $p_2$ are different primes, and $\sqrt{d} \...
1
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2answers
47 views

Prove that if $a$ is algebraic over $F$, then $F(a) = F(a^{-1})$.

Let $E$ be an extension of a field $F$ and let $a \in E$ be nonzero. Given that $a$ is algebraic over $F$ if and only if $a^{-1}$ is algebraic over $F$. Prove that if $a$ is algebraic over $F$, then $...
1
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1answer
38 views

Prove that $a$ is algebraic over $F$ if and only if $a^{-1}$ is algebraic over $F$.

Let $E$ be an extension of a field $F$ and let $a \in E$ be nonzero. Prove that $a$ is algebraic over $F$ if and only if $a^{-1}$ is algebraic over $F$. $\textbf{My Attempt:}$ prove of "if": ...
1
vote
1answer
63 views

Prove the minimal polynomial $a$ over $\mathbb{Q}$ is equal to the minimal polynomial $\overline{a}$ over $\mathbb{Q}$

Let $a \in \mathbb{C}$ be such that $a \notin \mathbb{R}$ and $a$ is algebraic over $\mathbb{Q}$. Let $\overline{a}$ be the complex conjugate of $a$. Let $f(x)$ be the minimal polynomial $a$ over $\...
2
votes
1answer
46 views

$F[x_1,…,x_n]$, $F(x_1,…,x_n)$, $F(\alpha_1, …, \alpha_n)$ and $F[\alpha_1, …, \alpha_n]$

I am having a hard time understanding the difference between $F[x_1,...,x_n]$ and $F[\alpha_1, ..., \alpha_n]$ (let $F$ be a subfield of a field extension $K$). If I understand this correctly, the ...
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votes
0answers
48 views

$[\mathbb{F}_{p^n}: \mathbb{F}_p] = n$?

I have seen multiple answers using this fact: $[\mathbb{F}_{p^n}: \mathbb{F}_p] = n$. Proving that $f(x)$ divides $x^{p^n} - x$ iff $\deg f(x)$ divides $n$ Prove that if $\mathbb{F}_{p^n} \subseteq \...
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0answers
37 views

Why is this extension of composite fields a Galois extension?

Let $k$ be a finite field and let $L/k$ be a a finitely generated field extension. Furthermore, let $K$ be a finite Galois extension of $L$ and let $k^\prime$ denote a finite intermediate field $k\...
2
votes
0answers
42 views

Let $F$ be a field with $|F|=q$ and $[K:F]=2$. Let $α∈F$ of order $q-1$. Then there exist an element $β \in K$ of order $q^2-1$ such that $β^{q+1}=α$

Let $F$ be a field with $|F|=q$ and $[K:F]=2$. Let $α∈F$ of order $q-1$. Then there exist an element $β \in K$ of order $q^2-1$ such that $β^{q+1}=α$ Am stuck with finding such a $\beta$ of order $q^...
1
vote
1answer
54 views

Calculate Galois group of finite degree extension of the rationals in GAP

is there a way to calculate the Galois group of finite field extension in GAP? I've tried the following but it doesn't work. ...
0
votes
0answers
19 views

$[B/\mathfrak{P}^n:A/p]=\sum_{k=0}^{n-1}[\mathfrak{P}^k/\mathfrak{P}^{k+1}:A/p]=n[B/\mathfrak{P}:A/p]$

If $A$ is a Dedekind domain with field of quotients $K$, $L$ is a finite separable extension of $K$ and $B$ is the integral closure of $A$ in $L$ then it is known that $B$ is also a Dedekind domain. ...
1
vote
1answer
38 views

Minimal Polynomial of $i\sqrt[4]{2}$ over $\mathbb{Q}(\sqrt[4]{2})$

I see that $$i\sqrt[4]{2} \notin \mathbb{Q}(\sqrt[4]{2})=\{a+b\sqrt[4]{2}:a,b\in \mathbb{Q}\}$$ And so the minimal polynomial has degree $\deg(m_{i\sqrt[4]{2}})\neq 1$. Let $\alpha=i\sqrt[4]{2}$, ...
2
votes
1answer
34 views

Questions about rings and ideals

Given a ring $R$ and an ideal $I \triangleleft R$ i'm trying to make some conclusions about the quotient $I/R$. Let for instance $R=\mathbb Q[x]$ and $I=(x^4-16)$. I'm looking for a zero divisor ...
0
votes
1answer
35 views

Questions about group extensions, and finding inverse elements

Let $I=(x^4+x+1)\vartriangleleft\mathbb F_2 [x]$ and $R = \mathbb F_2[x] / I$. I am to: List irreducible polynomials of $\mathbb F_2$ with degree smaller than 5. Show that $R$ is a field and find the ...

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