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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An field extension of degree 2 is Normal Extension.

let $L\ \text{be a field and $K$ is extension of $L$ such that $[K:L]=2$ prove that $K$is normal extension} $ what i have tried is let $ f(x)$ $\text{be any irreducible polynomail in} $ $L[x] $ ...
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Abstract regular curve over non-algebraically closed field

In Hartshorne chapter I.6 is discussed the construction of the abstract nonsingular curve as part of the proof for the well known correspondence between complete regular irreducible algrebaic curves ...
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22 views

Characterizing algebraic extensions

Show that $K$ is an algebraic extension over $F$ if and only if for every intermediate field $E$, every monomorphism $\sigma: E\rightarrow E$ which is the identity on $F$ is in fact an automorphism ...
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22 views

Simplifying Rational Expressions in a Finite Field Extension

In Dummit and Foote's textbook one of the exercises is: Let $\theta$ be a root of $x^3-2x-2$ over $\mathbb{Q}$. Compute $\frac{1+\theta}{1+\theta+\theta^2}$ in $\mathbb{Q}(\theta)$. My approach ...
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37 views

Algebraic extension of perfect field is algebraically closed

Let $F$ be a perfect field, i.e. every irreducible polynomial over $F$ has distinct roots in the algebraic closure of $F$. Suppose that $K$ is an algebraic extension of $F$ with the property that ...
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41 views

Proof of equality of $2$ polynomials without using Galois Theory?

I have the following situation and ask myself whether an argument is available to prove the equality of the $2$ following polynomials. Both polynomials are of degree $n$ and are monic. The ...
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30 views

If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?

If $\alpha$ is an algebraic element and $L \subset K$ are both field, does the polynomial ring $L[\alpha]$ is also a field? I am trying to prove that the ring of fraction $L(\alpha)$ is equal to ...
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Can I find a Galois extension which contains a finite set of algebraic elements?

Suppose $K/F$ is an algebraic extension of fields. Pick a finite collection $a_1,...,a_n \in K$. Can I find a Galois extension of $F$ which contains these $n$ elements of $K$?
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Is this inheritance valid for algebraic extensions?

Let $E/F$ be an algebraic extension. Let $\sigma:F\to E$ be a homomorphism in the category of fields. Can it be shown that the extension $E/\sigma(F)$ is also algebraic?
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Construct field Extension given that $\alpha$ has the following minimal polynomial

I apologize I have nothing to show that I have really attempted this question, but it's simply because I am struggling to get used to the terms and ideas. I randomly chose a question from a past paper ...
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3answers
41 views

Find the inverse formula of $p+qa+ra^2$

Where $a= \sqrt[3]{2}$. I need to find the inverse formula for numbers of the form $p+qa+ra^2$. So simply, $\frac{1}{p+qa+ra^2}$ but I assume I need to rearrange it to look "nicer"(It technically ...
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35 views

Splitting field.

Show that $\mathbb {Q }(\pi)$ is not splitting field over $\mathbb {Q }(\pi^2)$. I am thinking $\mathbb {Q }(\pi)$ and $\mathbb {Q }(\pi^2)$ are same field Or $\mathbb {Q }(\pi)$ is not even a ...
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2answers
48 views

Minimal Galois extension, describe structure of $Gal(L/\mathbb Q)$

Find the minimal Galois extension $L$ of $\mathbb Q$ containing $\mathbb Q(\sqrt[4]{5})$. Describe the structure of $Gal(L/\mathbb Q)$. I think $L$ is a splitting field of $X^4-5$ over $\mathbb Q$. ...
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52 views

Is the tensor product of 2 finite extension of $\Bbb Q$ isomorphic to a direct sum of fields?

I have $K_1$ and $K_2$ two finite extensions of $\Bbb Q$. I can construct $K_1 \otimes_\Bbb Q K_2$. This is clearly isomorphic to a direct sum of field as vector space (indeed one can easily see that ...
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1answer
34 views

Splitting field of $x^4-4x^3+4x^2-3$

I've got that $x^4-4x^3+4x^2-3 =(x^2-2x+ \sqrt{3})(x^2-2x-\sqrt{3})$ The roots of the polynomials are: $\alpha = 1+\sqrt{1-\sqrt{3}}$ $\quad$ $\alpha_1= 1-\sqrt{1-\sqrt{3}}$ $\quad$ $\beta= ...
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2answers
34 views

Completion of an extension $K|\mathbb{Q}$ w.r.t. a non archimedean abs. value, isomorphic to $K\cdot \mathbb{Q}_p$

As the title suggests I want to prove the following result: given $\mathfrak{p}|(p)$, prime ideal of $\mathcal{O}_K$ over $p$, we have the canonical absolute value induced by it on $K$, and we can ...
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1answer
59 views

Can you construct a field of characteristic $\neq 0, 2$ such that every one of its subrings is also a field?

A friend asked me this a few days ago, and I was thinking that it may be impossible, but now I'm not so sure. He suggested a "nonprincipal ultrapower" $(\mathbb{Z}/(2))^{N}$ such that every subring is ...
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2answers
51 views

Determine the degree of the extension $\mathbb{Q}(\sqrt{3 + 2\sqrt{2}})$.

The following is from a set of exercises and solutions. Determine the degree of the extension $\mathbb{Q}(\sqrt{3 + 2\sqrt{2}})$ over $\mathbb Q$. The solution says that the degree is $2$ since ...
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1answer
32 views

Inseparable extensions

Given that $F/K$ is a finite extension of fields and the extension is not separable. My question is whether we can always find a subfield $L$ such that $K\subset L \subset F$ and $[F:L]=p$, where ...
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1answer
17 views

How can we find $[GF(p^n):GF(p)]=n$?

I was searching why $[GF(p^n):GF(p)]=n$. It is not very logical, isn't it ? I know that $$GF(p^n)=\{x\in GF(p)^{alg}\mid x^{p^n}=x \}$$ is a field with $p^n$ element since it split $X^{p^n}-X$ which ...
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25 views

One point compactification and field extension

We know that a topological space $X$ has a one point compactification if there exist a compact set $Y$ having $X$ as a subspace,where $Y\setminus X$ contains a single point Also for two such $Y$ and ...
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1answer
43 views

Why does an algebraically closed field not have any non-trivial algebraic field extensions?

Let $K$ be an algebraically closed field. Then there are no non-trivial algebraic field extensions of $K$. I can understand that if the field extension is of the form $K[x]/\langle p(x)\rangle$, ...
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48 views

The Galois Field for the polynomial $x^3 - 2$.

I am reading a textbook prior to taking my first course in Field Theory. I think that if someone could answer the following 4 questions simply as True of False I might be less confused. I am denoting ...
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26 views

Splitting field of $x^2 +1 \;$ over $\mathbb Z_7 $

I need to find the splitting field of $\; x^2+1 \in \mathbb Z_7 [x] \;$ over $\mathbb Z_7 $. The roots of the polynomial are $-i \;$ and $i$. Therefore I would conclude that the splitting field is ...
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3answers
29 views

Degree of a field extension with three elements

I am struggling with the following: Let's have the field extension $R/Q$. How can I find $[Q(\sqrt 6,\sqrt 10,\sqrt 15):Q]$? (*) So far all I have is that: $[Q(\sqrt 6):Q] = 2$, because ...
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1answer
28 views

Show that the set {${\alpha \in F_{5^6} | F_5(\alpha)=F_{5^6} }$} contains 15480 elements

Question: Show that the set {${\alpha \in F_{5^6} | F_5(\alpha)=F_{5^6} }$} contains $15480$ elements Since this number is so large, I think there is some trick to get the answer. Also $15480$ is ...
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2answers
90 views

General technique for finding minimal polynomial? [closed]

I always have a lot of trouble with these problems, "find the minimal polynomial of {number} over {field}" What are the general procedures for solving problems of this format? Thank you for your ...
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29 views

Show that a finite field of order $p^n$ has exactly one subfield of $p^m$ elements for each divisor $m$ of $n$.

Show that a finite field of order $p^n$ has exactly one subfield of $p^m$ elements for each divisor $m$ of $n$. Suppose that $o(F)=p^n$ .Let $F$ has $\Bbb Z_p$ as its prime subfield. Let $n=km$. I ...
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1answer
20 views

Prove that $\Bbb Z_2(\alpha)=\Bbb Z_2(\beta)$ .

Consider the polynomials $x^3+x^2+1,x^3+x+1$ over $\Bbb Z_2$ which have roots say $\alpha,\beta $ respectively. Prove that $\Bbb Z_2(\alpha)=\Bbb Z_2(\beta)$ . Since both $x^3+x^2+1,x^3+x+1$ are ...
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1answer
39 views

Prove that every finite extension field of $\Bbb R$ is either $\Bbb R$ itself or isomorphic to $\Bbb C$.

Prove that every finite extension field of $\Bbb R$ is either $\Bbb R$ itself or isomorphic to $\Bbb C$. I tried in this way.Let $E$ be a a finite extension of $\Bbb R$. Then $E$ is an ...
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1answer
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Does $S = R \cap K$ of a field extension $K \subseteq L = Q(R)$ satisfy $Q(S) = K$?

If $K$ is finite field, then one can easily show that there is no proper subring $R$ with $Q(R) = K$, where $Q(R)$ is the field of fractions of $R$. As a consequence, algebraic extensions $K$ of ...
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1answer
58 views

Show that the only subfields of $\mathbb{Q}(i, \sqrt{5})$ is $\mathbb{Q}, \mathbb{Q}(i),\mathbb{Q}(\sqrt{5}), \mathbb{Q}(i \sqrt{5})$ and itself?

I'm reading Stewart's Galois Theory and encountered this exercise in Chapter 8. I want to show this by contradiction: Assuming there exists a proper subfield $\mathbb{Q}(\alpha)$ of $\mathbb{Q}(i, ...
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1answer
55 views

If $\sqrt[3]{2}+\omega\in \Bbb{K}$, prove that $\sqrt[3]{2}\in \Bbb{K}$

Let $K=\Bbb{Q}(\sqrt[3]{2}+\omega)$ be a field extension of $\Bbb{Q}$ where $\omega$ is a root of $x^2+x+1$. Prove that $\sqrt[3]{2}\in K$ How do I prove this? I am at a loss. I did it by a very ...
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1answer
36 views

$\mathbb{Q}$-automorphisms of $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$

I'm reading Stewart's Galois Theory. In Chapter 8, we define that: Let $L:K$ be a field extension, so that $K$ is a subfield of the subfield $L$ of $\mathbb{C}$. A $K$-automorphism of $L$ is an ...
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25 views

Which of the following field properties are correct?

Let $\omega = \cos{\frac{2\pi}{10}}+i\sin{\frac{2\pi}{10}}$. Let $K = \mathbb{Q}(\omega^2)$ and $L = \mathbb{Q}(\omega)$. Then $[L : \mathbb{Q}] = 10.$ $[L : K] = 2$. $[K : ...
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64 views

If b is algebraic over F(a), and a is algebraic over F, then is b algebraic over F?

If $b$ is algebraic over the field $F(a)$ then is it algebraic over the field F? I would like to find a proof if it is true or a counter example if it is not. The only thing I could think of was ...
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1answer
35 views

Simple extension of a countable field is also countable?

How do we know "Simple extension of a countable field is also countable?" This intuitively makes sense but I'm not sure how to rigorously prove this. I want to use this to prove that there does not ...
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Proving that isomorphism of field extension is an equivalence relation

I'm reading Stewart's Galois Theory Third Edition. In Chapter 4 he gives the definition of the isomorphism of field extension as follows: An isomorphism between two field extension $l : K ...
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find all composite order fields between 200 and 900 . [closed]

There are many fields of composite order between 200 and 900 how can I find those fields.
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81 views

The maximal subfield of $\mathbb C$ not containing $\sqrt2$

Related: Does a maximal subfield of $\mathbb C$ not containing $\sqrt{2}$ have index $2$? He said, "...fixed field is an extension of $K$ which doesn't contain $\sqrt{2}$, and thus must be $K$ ...
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1answer
23 views

Prove that if $\mathbb{F}_{p^n} \subseteq \mathbb{F}_{p^m}$ then $n \mid m$

The order of $\mathbb{F}_{p^n}$ is $p^n$ and $\mathbb{F}_{p^m}$ is $p^m$. Since $\mathbb{F}_{p^m}$ is the largest field and it contains all the elements of $\mathbb{F}_{p^n}$ and the order of any ...
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37 views

possible degrees of the minimal polynomial of t over F [closed]

Please help me with this Abstract Algebra question: Let $F \subset K$ be a Galois extension with Galois group isomorphic to the alternating group $A_4$. Let $t \in K$ be an element. Determine the ...
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1answer
20 views

$K(ab,a+b) \subset K(a,b)\;$ finite field extension

Let $\; K(ab,a+b) \subset K(a,b)\subset L \quad a,b \in L$ Is $\; K(ab,a+b) \subset K(a,b) \;$ a finite field extension and if not can anyone give a counterexample ?
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60 views

Prove that $\sqrt{2} \notin \mathbb{Q}(\sqrt{3})$.

Suppose there exists an isomorphism $\Phi \colon \mathbb{Q}(\sqrt{2}) \to \mathbb{Q}(\sqrt{3})$. Then, of course, it must be the case that $\Phi(1) = 1$. Hence \begin{align*} 2 &= 1+1 = \Phi(1) + ...
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26 views

Subfield of $\mathbb C$ with certain degree.

Let $n\in\mathbb N$. then, is there a subfield of $\mathbb C$ such that $[\mathbb C:K]$= $n$ ? if $n$ = 1,2, then answer is yes. But I don't know if $n$ is larger than 2. How do I proceed more? ...
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29 views

Galois extension of $\mathbb C$ is also Galois over $\mathbb R$??

Is any Galois extension of $\mathbb C$ also Galois over $\mathbb R$? I know if that extension is finite, Then it is true because $\mathbb C$ is algebraically closed. But How about the infinite case? ...
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1answer
53 views

If degree of extension is infinite then intermediate ring not need to be a field. [closed]

Let $F\subset K$ be a field extension and $D$ be an intermediate ring such that $F\subset D\subset K$. If $[K:F]$ is infinite then $D$ is not necessary a field. So basically I need a counter ...
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217 views

$\sqrt[31]{12} +\sqrt[12]{31}$ is irrational

Prove that $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational. I would assume that $\sqrt[31]{12} +\sqrt[12]{31}$ is rational and try to find a contradiction. However, I don't know where to start. Can ...
5
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1answer
59 views

Easy criteria to determine isomorphism of fields?

Let $K$ be a field and $f,g$ irreducible polynomials in $K[X]$, is there a nice iff condition for $K[X]/(f)\cong K[X]/(g)$? ($\cong$ denotes an isomorphism that is the identity on restriction to ...
2
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2answers
46 views

Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable?

Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable ? I would say yes since the fact that $x$ separable over $F$ implies $E(x)/F$ separable, an since $E=E(x)$ then $E/F$ ...