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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Factoring polynomials modulo 3

Let $f(x) = x^5 + 2x^2 + 2x + 2 \in\mathbb Z_3[x]$. Then the irreducible factorization of $f(x)$ is $(x^2 +1)(x^3+2x+2)$ even though it does not have a root in $\mathbb Z_3$. How did we find that ...
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A lemma about the Galois group of radical extensions in Rotman's book Advanced Modern Algebra

Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. (What he calls a 'pure extension' is commonly called 'radical extension' by most authors.) I am confused by ...
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If a sequence is generated by a $\mathbb{Q}$-polynomial passed mod $p$, can we find an appropriate polynomial over an extension of $\mathbb{F}_{p}$?

If we have a polynomial that takes integer values for integer inputs, we can take its outputs at integer inputs and pass them $\text{mod }p$. However, my understanding is that the coefficients of the ...
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Lemma A-5.19 about the Galois group of radical extensions in Rotman's book Advanced Modern Algebra

Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. It is about the characterization of the Galois group of pure extensions (which are mostly called radical ...
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Clarification about field extension and its degree

I know there are some posts about this, but I'm still confused regarding this specific question. It is said that the dimension of any field extension $\mathbb{Q}(w)$ is the degree of the irreducible ...
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non isomorphic algebraically closed fields

I am trying to find all algebraically closed fields(up to isomorphism). I found that the field of all algebraic numbers over $\mathbb Q$ is algebraically closed and I also know that the field of ...
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Fraction field and Monic Polynomial

Here is the question. $R$ is a UFD and $F$=Frac$R$, $E=F(α)$ is an algebraic extension of $F$. Prove of disprove that, if the minimal polynomial of $α$ over $F$ belongs to $R[x]$, then $α$ is a root ...
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Vector Space and Expansion of Fields

I have learnt the complexification of real vector spaces. Let $V$ be a vector space over $\mathbb{R}$, and we can define a new vector space $V_{\mathbb{C}}$ which is a complex vector space and is ...
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Extension degree related to a transcendental extension in positive characteristic

Let $k$ be algebraically closed field of characteristic $p > 0$. Let $k(t)$ be a transcendental extension. Let $L$ be the splitting field of $x^n-t$ over $K$ $(n \geq 1)$. Let $G = Aut(L/K)$ and $F ...
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Tower laws, intersection and product of fields

I am working on a problem sheet question of Galois theory where I get stuck: Assume all field extensions here are finite. Consider subfields $E$ and $F$ of $\Omega$. Let $EF$ denote the smallest ...
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Is the following statement about splitting fields true?

Let us take a look at the following definitions: Let $F$ be a field and $f(x)\in F[x]$ then a field extension $E$ of $F$ is said to be the splitting field of $f$ over $F$ if $f$ splits completely in $...
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How can I show that $\mathbb{Q}(\alpha^3)\subsetneq\mathbb{Q}(\alpha^3\sqrt{2})$?

I am struggling with this field extensions problem. Let $\alpha \in \mathbb{C}$ be a root of $x^5+7x^2-14x+14 \in \mathbb{Q}[X]$. Show whether the following statement is true or false: $$ \mathbb{Q}(\...
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Equivalent definitions of Galois extension.

I am studying Galois theory.I encountered some equivalent definitions of Galois extension: The following are equivalent for a finite extension: $1. E/K$ is a splitting field of separable polynomial ...
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Are there any interpretations of field theory which would allow for a negative degree of a field extension?

I've only ever seen finite field extensions indicated as $[L:K] < \infty$. I've never seen $-\infty<[L:K]<\infty$. I take this to mean that field extensions of a negative degrees are not ...
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What is the lattice diagram of the Galois extension $\mathbb{Q}(\sqrt 2, \sqrt 3, \sqrt{(2+\sqrt 2)(3+\sqrt 3)})/\mathbb{Q}$?

Consider the Galois extension $K:=\mathbb{Q}(\sqrt 2, \sqrt 3, \sqrt{(2+\sqrt 2)(3+\sqrt 3)})/\mathbb{Q}$. As it is clear, it is of degree $8$ extension. So the Galois group is a group of order $8$. I ...
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Field extension of union of sets

Basically there is to prove that for e field extension $E/K$ and $S,T\subseteq E$ subsets, then $K(S)(T)=K(S\cup T)$. I proved it in this way; $K(S\cup T)$ is a field which contains S, then $K(S)\...
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Determine the basis of a composite field extension using the primitive element

I have been confusing myself a lot with the following and I am sure I must be missing something obvious, so sorry for this probably stupid question. Given $\alpha = \sqrt{2} - \sqrt{3}$ and $\beta = \...
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Degree of elements in simple extension of $\mathbb{Q}$

Is the following statement true? Let $x$ be algebraic over $\mathbb{Q}$ and let degree of $x$ over $\mathbb{Q}$ be n. Then, simple extension $\mathbb{Q}(x)$ will be equal to the set: $$\{a_0 + a_1x + \...
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Finding subfields of an Extension Field [duplicate]

Let a $\in$ C be a root of the polynomial $X^4+ 1 \in Q[X]$ Consider the field extension Q(a) of Q. Find three fields $K_1, K_2,K_3$ such that $Q \subset K_i \subset Q(a)$ for i=1,2,3. I found out 2 ...
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2 votes
1 answer
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Degree of subfield fixed by single automorphism

Let $L/K$ be a finite Galois extension. If $\sigma \in \mathrm{Gal}(L/K)$ has order $d$, is it the case that $$L^\sigma := \{ \ell \in L : \sigma(\ell) = \ell\}$$ satisfies $[L^\sigma:K] = d$? This ...
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Given $K/F$ where $K$ is a field extension of $F$ of the form $F[x]/\langle p(x) \rangle$, what is the structure of $K[x]$?

$F$ is a field. $\langle p(x) \rangle$ is a maximal ideal. So $K = F[x]/\langle p(x) \rangle$ is a field extension. I am trying to understand what would be the structure of $K[x]$? $F[x]$ has all ...
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What are the $\Bbb Q$-automorphisms of $\Bbb Q(\alpha)$ with $\alpha$ a root of $x^3-3x^2+3$?

I know the polynomial is irreducible over $\Bbb Q$ and the roots are real irrational. A $\Bbb Q$-automorphism $\sigma:\Bbb Q(\alpha)\to \Bbb Q(\alpha)$ is determined by $\sigma(\alpha)$, which is also ...
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Is it possible to produce identically-behaving binary extension fields using different irreducible polynomials?

Let $GF(2^m)$ be a binary extension field with constructing polynomial $f(z)$ be an irreducible, primitive polynomial over $GF(2)$. Is there any possibility that two (or more) different $f(z)$ can ...
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1 answer
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Question regarding Galois group of a polynomial.

I am a graduate student.We have Galois theory in this semester.We were first taught splitting fields of a polynomial.Then our instructor introduced the Galois group of a polynomial $f\in F[x]$ to be $...
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There are uncountably many subgroups of $\text{Gal}( \bar{\mathbb{Q}}/\mathbb{Q})$ of index $2$.

I am reading infinite Galois theory, and the motivation for introducing the Krull topology seems to be that this is the way in which we can solve the problem that the Galois correspondence fails. This ...
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Extensions by Adjoining elements and Extensions by quotient of a Principal Ideal

Extensions can be constructed 2 ways to get an extension with roots of a polynomial Adjoining an element to a field - i.e. $F(\sqrt 2)$ is an extension of $F$. You can also build a tower of ...
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Primes induced by embeddings into completion

Asking myself how to construct extensions of fields where primes split completely, I came across this answer. I have problems to understand one part of the proof of the Lemma in the answer. Let's ...
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Proving that splitting fields are normal extensions

I am trying to prove that if $f \in K[x]$ has splitting field $L$ over $K$, then $L/K$ is normal (i.e. if $g \in K[x]$ irreducible and has a root $\alpha \in L$ then all of the roots of $g$ lie in $L$....
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Consider the following extensions of $\mathbb{Q}$: $\mathbb{Q}(1+\sqrt[3]{5})$ and $\mathbb{Q}(\sqrt{7}, i)$.

Which of them are separable and which are normal? That is the question and my respective answers follow below. I hope you have suggestions or if there are any mistakes I hope you help me fix them. ...
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Consider $E = \mathbb{Q}(\sqrt{2}, \sqrt{7} )$.

Here are the following questions and my respective answers follow below. I hope you have suggestions or if there are any mistakes I hope you help me fix them. Find a basis for $E$ over $\mathbb{Q}(\...
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Are field isomorphisms necessarily linear isomorphisms?

Let’s say we have a finite field extension $L/K$. Suppose we have that $L(\alpha )$ and $L(\beta )$ are isomorphic as fields (finite extensions) with an isomorphism $\phi $. Now viewing $L(\alpha ) $ ...
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3 votes
1 answer
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Find the units in the ring of integers over $\mathbb{Q}(e^{i\pi/6})$

Let $\omega=e^{i\pi/6}$, $L=\mathbb{Q}(\omega)$ and $O_L$ the ring of integers of $L$. Let $K=\mathbb{Q}(\sqrt{3})$ and $O_K$ the ring of integers in $K$. Show that the units in $O_L$ are $O_L^{\times}...
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Proof checking - Galois theory

I am trying to show that if $K$ is a field and $f \in K[x] $ has exactly $n$ distinct roots, say $\alpha_1 ,..., \alpha_n \in L $ where $L$ is a splitting field (so $L=K(\alpha_1 , ..., \alpha_n ) $) ...
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Algebraic extension $K\subset F$ containing (splitting field)/(root) of any $P\in K[x]$ is algebraically closed. Where is my mistake?

I have the following problem: $\textbf{(i)}$ Let $K \subset F$ be an algebraic extension that contains splitting field of any polynomial $P\in K[x].$ Prove that $F$ is algebraically closed. $\textbf{...
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Every number field of degree $3$ is of the form $\mathbb{Q}(\alpha)$ where $\alpha$ has minimal polynomial $x^3 + ax + b$

I want to show that every number field of degree $3$ is of the form $\mathbb{Q}(\alpha)$ where the minimal polynomial of $\alpha$ is of the form $m_\alpha(x) = x^3 + ax + b$. Let $K$ be the cubic ...
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differential extensions meaning, use, validity of some expressions, derivation, := meaning

In the attached image IK means field characteristic zero, [...] means polynomial and IK[x,y] means polynomial in x and y in IK and $m(x,y)$ is generally set to $0$. If you need more detail or to see ...
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1 answer
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If $F$ is algebraic over $K$ and $D$ is an intermediate and an integral domain, is $D$ a field?

I would imagine it would be as simple as showing that $D$ is finite since a finite integral domain is a field. I know that if $F$ is a finite extension of $K$, then $F$ is an algebraic extension of $K$...
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2 votes
1 answer
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Find a particular intermediate field $M$ such that $\mathbb{Q}\subset M\subset\mathbb{Q}(\sqrt{\frac{3+i\sqrt{7}}{2}})$.

Problem: Let $\alpha=\sqrt{\frac{3+i\sqrt{7}}{2}}$ and $K=\mathbb{Q}(\alpha)$. Find the fixed field $M=\{x\in \mathbb{Q}(\sqrt{\frac{3+i\sqrt{7}}{2}})|\sigma(x)=x\}$, where $\sigma$ is the ...
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2 votes
2 answers
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Question on the irreducibility of $x^4 - 10x^2 +1$

I am currently doing some work in Galois theory, and the following situation has me perplexed. The polynomial $~x^4 -10x^2 + 1 $ is irreducible over $\mathbb Q, $ which can be shown in a variety of ...
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L is normal extension if it's splitting field

I am trying to understand this proof That if L is splitting field of some polynomial then it's normal extension . I got the part till we get that $L(\alpha)/K(\alpha)$ and $L(\beta )/K(\beta)$ are ...
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Inequality involving degrees of field extensions

Suppose we have fields $F \subset E \subset K$ and $\alpha \in K$. Suppose also that $K$ is algebraic over $E$. Let $\alpha \in K$. Then we have $p(\alpha)=0$ for some $p(x) = a_{0} + a_{1}x+ \dots+a_{...
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1 vote
1 answer
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Proof of If $K$ is a finite extension of $F$, then $K$ is algebraic and finitely generated over $F$

The following lemma appears in Fields and Galois theory book by Patrick Morandi If $K$ is a finite extension of $F$, then $K$ is algebraic and finitely generated over $F$ and in its proof uses the ...
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Why $\Bbb{F}_{p^n}((t))/ \Bbb{F}_p((t))$ is finite extension?

Let $p$ be a prime. Why $\Bbb{F}_{p^n}((t))/ \Bbb{F}_p((t))$ is finite extension ? I tried to prove this by natural inclusion, $\Bbb{F}_p((t))$→$\Bbb{F}_{p^n}((t))$'s kenel is degree $n$ irreducible ...
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What is the practical use of constructing an extension $\mathbb R[x]/\langle p(x) \rangle$ to find a root rather going to $\mathbb C$?

If you have a polynomial (say $p(x)$) which is irreducible in $\mathbb R[x]$, then is there a practical use of constructing an extension field $\mathbb R[x]/\langle p(x) \rangle$ with a zero rather ...
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$\operatorname{dim}_{\mathbb C} \mathbb C \otimes_{\mathbb R} \mathbb C$ vs. $\operatorname{dim}_{\mathbb R} \mathbb C \otimes_{\mathbb R} \mathbb C$

Here is the spaces I am trying to find their dimensions: $1- \operatorname{dim}_{\mathbb C}(\mathbb C \otimes_{\mathbb R} \mathbb C)$ $2-\operatorname{dim}_{\mathbb R}(\mathbb C \otimes_{\mathbb R} \...
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3 votes
2 answers
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Possible $|F(\alpha, \beta):F|$ where $|F(\alpha):F|=6$ and $|F(\beta):F|=15$?

Here is a past paper problem which I am struggling to solve currently. Let $\alpha,\beta\in E$, where $E$ an extension of field $F$. We are given $|F(\alpha):F|=6$ and $|F(\beta):F|=15$. What are the ...
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1 answer
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Irreducible polynomial in integers modulo p

I am a completing a past paper question and I am undecided on what method to use here. The question is: For what $a$ is $f(x)=x^3+x+a\in\mathbb{Z}_{7}[x]$ irreducible? My ideas are: (1) Check each $a\...
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1 vote
1 answer
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Is $\mathbb{Q}(\sqrt[3]{3},\sqrt{3},i)$ a splitting field over $\mathbb{Q}?$

Is $\mathbb{Q}(\sqrt[3]{3},\sqrt{3},i)$ a splitting field over $\mathbb{Q}$? I am thinking that if i prove $\mathbb{Q}(\sqrt{3})$ is a splitting field over $\mathbb{Q}$ and $\mathbb{Q}(\sqrt[3]{3},i)...
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Global sections are finite product of global sections of irreducible comoonents

I would like to extend the question here. Assume that $X\rightarrow Spec(K)$ is proper and X is reduced. Since is proper and $Spec(K)$ is Noetherian, $X$ is Noetherian and hence it will be finite ...
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Simplify the splitting field of $t^6-2t^3-1$ over $\mathbb{Q}$. [duplicate]

Problem: Simplify the splitting field of $t^6-2t^3-1$ over $\mathbb{Q}$. My Attempt: The roots for $f(t)=t^6-2t^3-1\; \text{in}\; \mathbb{C}$ are $\{\alpha, \beta, \zeta\alpha, \zeta\beta, \zeta^2\...
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