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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field extension $F\subset E$ with both separable and inseparable elements

Can someone please give me an example of a field extension $F\subset E$ such that E\F has both separable and inseparable elements? if $F(\alpha)$ is a simple extension of F, and if $\alpha$ is ...
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27 views

what is the minimal condition for two elements to create same field extension?

Given a field $K\subset E$, with $\alpha,\beta\in E$, such that $K(\alpha)=K(\beta)$. What can we then say about $\alpha$ and $\beta$? If the extension is finite, then $\alpha$ is a linear ...
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0answers
14 views

for which $\alpha \in K$ $det(xI-L_{\alpha})=min(F,\alpha)$

Let $K $ and $F$ are two fields, $K=F(a)$ suppose $[K:F]=n$ for $\alpha \in K$ let $L_{\alpha}$ be the $F$ linear transformation $K$ to $K$ defined by $L_{\alpha}(x)=\alpha x$. Now my question is for ...
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1answer
29 views

Existence of an intermediate field $K\subseteq M \subseteq L$ such that $[L:M]=p$

Let $L/K$ be a finite galois extension (normal and separable). Let $p$ be a prime number which divides $[L:K]$. Is there necessarily an intermediate field $K\subseteq M \subseteq L$ such that ...
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27 views

What's so special about quadratic extensions?

Reading through chapter 13 "Field Theory" from Dummit and Foote Algebra. I am wondering why such an emphasis is placed upon "quadratic extensions" of a field F. They state that for any field F ...
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1answer
25 views

A Field Extension of a Cyclic Galois Group is Galois

Let $F \subseteq E $ be extension of fields. If $Gal(E/F)$ is a cyclic group, does it imply that the extension $E/F$ is a Galois extension? If not, any example?
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1answer
10 views

Degree of a splitting field of a polynomial

If $F \subseteq E$ is a field extension, and $E$ is a splitting field of a polynomial $f(x)\in F[x]$ of degree n over $F$. Does that mean $|E:F|\le n$ ? Could we conclude something more? If someone ...
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31 views

Complete field and field extension.

$(K,u)$ be a pair of the field $K$ and its absolute value $u$, $(K_u, \bar u)$ denotes its completion and the corresponding absolute value. Let $L$ be a field containing $K$, $\pi:K_u\rightarrow ...
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1answer
43 views

Quadratic field extensions and complex conjugation

If you consider any quadratic extension $K$ of $\mathbb{Q}$, it has to be fixed by complex conjugation, because from $[K : \mathbb{Q}] = 2$ we know $K | \mathbb{Q}$ has to be a normal extension and as ...
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0answers
30 views

Unique extension of the absolute value

Let $(K,u)$ be a complete valued field, $u$ be its discrete absolute value (corresponds to a discrete valuation on $K$), then: ($\ast)$ Let $E/K$ is a finite separable field extension, then the ...
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1answer
34 views

Question about the splitting field of a finite separable extension

Let $k_s|K|k$ be a tower of field extensions and $K|k$ be finite and separable ($k_s$ is a separable closure of $k$). There exists $\alpha\in K$ such that $K=k[\alpha]$. Then the splitting field $L$ ...
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1answer
31 views

If $E/F$ is finite divisible, then it is separable.

I want build a separable extension $E/F$. Suppose that $E/F$ is a finite divisible field extension. I want to prove that $E/F$ is separable in this method: we know that if $Char(F)=0$, then $E/F$ is ...
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1answer
39 views

Fraction field of $F[X,Y](f)$ isomorphic to $F(X)[Y]/(f)$

Assume $F$ is a field and $f$ is an irreducible polynomial in $F[X,Y]$ which involves the variable $Y$. Then, by Gauss's lemma, $f$ is irreducible also in $F(X)[Y]$ so that $F(X)[Y]/(f)$ is a field ...
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unramified quadratic extension of number field

I try to understand the following statement There are only finitely many quadratic unramified extension of a number field $K$ I know by Kummer theory that such extensions are of the form ...
2
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3answers
73 views

Does $\cos (\pi/5)$ belong to $\mathbb{Q} (\sin(\pi/5))$?

I need to know if $$\cos(\pi/5) \in \mathbb{Q} (\sin(\pi/5))?$$ I can compute explicitly such $\cos$ and $\sin$, but I have some difficulties how to deduce from this an answer.
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Prove that $R$ is also a field. [closed]

Let $K$ be an algebraic extension of $F$ and let $R$ be a subring of $K$ and $F \subseteq R \subseteq K$. Then prove that $R$ is also a field.
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1answer
23 views

Prove that $E=F[\alpha^2]$ [duplicate]

Let $E=F[\alpha]$, $\alpha$ is algebraic over $F$ and $[E:F]$ is odd. Prove that $E=F[\alpha^2]$. Now clearly $[F[\alpha^2]:F]|[E:F]$ so $[F[\alpha^2]:F]$ is also odd. But how can we show that ...
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1answer
55 views

Adjoining all roots of unity to an arbitrary field $F$, is an abelian extension?

I want to build an abelian extension of an arbitrary field $F$, such that any polynomial $x^n-1$ for all $n\in \mathbb{N}$ has an answer in it. So I want to adjoin all roots of unity to $F$. But I ...
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1answer
28 views

A question regarding proving the fact that every finite field is perfect

I am trying to prove the fact that every finite field is perfect. Hence, every irreducible polynomial is separable (does not have a repeated root). This is easy to prove when in a field of ...
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1answer
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Show that $\mathbb{Q}(\sqrt{2},\sqrt{3},\dots,\sqrt{p},\dots)$ is an algebraic extension of $\mathbb{Q}$, for $p$ prime.

I have shown that $[\mathbb{Q}(\sqrt{2},\dots,\sqrt{p},\dots):\mathbb{Q}]=\infty$, by showing that $$ \mathbb{Q} \subset \mathbb{Q}(\sqrt{2})\subset \mathbb{Q}(\sqrt{2},\sqrt{3})\ldots$$ is an ...
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3answers
33 views

Normal extension of $\mathbb{Q}(1-2i\sqrt2)/\mathbb{Q}$

How I can examinate if the extension $\mathbb{Q}(1-2i\sqrt2)/ \mathbb{Q}$ is normal? Could anyone give any hints for this?
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3answers
84 views

Showing $5^{1/3}$ is not in the field $\mathbb{Q}(2^{1/3})$ [duplicate]

I want to show $5^{1/3}$ is not in the field $\mathbb{Q}(2^{1/3})$. My reasoning was $[\mathbb{Q}(2^{1/3}):\mathbb{Q}]=3$ and $[\mathbb{Q}(5^{1/3}):\mathbb{Q}]=3$. Similarly we know that ...
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1answer
35 views

Finite fields - quadratic extensions

Let $F_q$ be a finite field of $\operatorname{char}F\neq2$. suppose $a_1$,$a_2$,...,$a_q$ are the elements in $F_q$. show there exists $i$ such that for any $j$ $a_i\neq a_j^2$. compute the degree ...
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0answers
23 views

Trace and Determinant of Field Extension

In algebra, we had a look at the trace and the determinant of a field extension. I am familiar with those concepts in linear algebra and I have seen that finite extensions can be viewed as a finite ...
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0answers
37 views

Irreducible polynomial over a perfect field of characteristic $p\neq 0$

Let $E/F$ be an algebraic field extension where $F$ is of characteristic $p\neq 0$. Let $F'=\{x\in E : x^{p^n}\in F \;\text{for some}\;n\geq 0\}$. Then $F'$ is a perfect field containing $F$ and $E$ ...
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2answers
39 views

Finding dimension of a field extension

How would anyone go about this problem? Find dim$_\mathbb{Q}\mathbb{Q}(\alpha,\beta)$ where $\alpha^{3}=2$ and $\beta^{2}=2$. Thanks for your help, I really don't know how to go about this ...
3
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0answers
24 views

Show that every algebraic field extension has a normal closure

I am trying to show that every algebraic extension of fields has a normal closure. Let $L/K$ be an algebraic extension. First suppose that the extension is finite. So $L=K(\alpha_1,...,\alpha_n)$ ...
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2answers
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Cyclotomic extension of $K$, $Gal_{K}{F}$ is isomorphic to a subgroup of $\mathbb Z_n^*$

Let $n$ be a positive integer, $K$ be a field with $charK$ does not divide $n$, and $F$ be the cyclotomic extension of $K$ of order $n$. Theorem says that $Gal_{K}{F}$ is isomorphic to a subgroup of ...
3
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1answer
36 views

Is $K := \mathbb{Q}(\cos (2\pi / 11))$ a Galois extension over $\mathbb{Q}$?

I believe that it is because $\cos(2\pi / 11) = (\zeta + \zeta^{-1})/2$ where $\zeta = e^{2\pi i/11}$ is a primitive $11$-th root of unity, and so $K$ is a subfield of $\mathbb{Q}(\zeta)$ with ...
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1answer
26 views

Galois group of maximally tamily ramified extension over the maximally unramified extension of a global function field F

Suppose $F$ were a local nonarchimedean field of characteristic zero of residue characteristic $p$. Let $F^{un}$ the maximally unramified extension. Let $F^{un}\subset F^{tame}$ be the maximally tame ...
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Verify that $\sqrt{1+\sqrt2+\sqrt3+\sqrt5}$ is constructible by determining the sequence of its extension fields

Verify that the following numbers are constructible by determining the sequence of their extension fields: $$ \sqrt{1+\sqrt2+\sqrt3+\sqrt5}, ...
3
votes
1answer
148 views

Can extension by an isomorphic field be of degree at least 2?

Suppose $K/F$ is a field extension such that $K\not=F$. Is it legitimate to say that $F$ and $K$ can't be isomorphic since by assumption \begin{equation*}[K:F]\ge 2\end{equation*}and if $K$ and $F$ ...
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1answer
27 views

A question about fields and separability in Serre's “Local Fields”

On page 14 of the English edition of Serre's "Local Fields", that is chapter 1, section 4, I am confused by the following; there is talk of fields $B/\mathfrak P$ and $A/\mathfrak p$ for prime ideals ...
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27 views

Choice of Primitive Element in ''Primitive Element Theorem''

Let $F$ be a field of characteristic $0$, and $K$ a finite extension of $F$. Then it is well known (see this) that $K$ can be obtained by attaching an element $\alpha \in K$ to $F$, i.e. ...
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2answers
67 views

If $X^p-a$ has no zeros in a field $F$ of characteristic $p$ where $a \in F$, is it irreducible?

Let $F$ be a field of characteristic $p>0$ and $a\in F$. I have an easy question which I'm stuck on. If the polynomial $X^p-a$ has no zeros in $F$ then is it irreducible over $F$? ...
2
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1answer
35 views

Factoring ideals in algebraic number rings using Dedekind's theorem

Let $K \subset L=K(\alpha)$ be a number field extension with rings of integers $\mathcal{O}_K$ and $\mathcal{O}_L=\mathcal{O}_K[\alpha]$ respectively. Let $\pi$ be a prime ideal in $O_K$, and let $F = ...
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0answers
42 views

Ordered fields are real?

This question is motivated by another question posted earlier today. Let $F$ be a field endowed with an embedding $F\hookrightarrow\Bbb R$. Then the ordering of $\Bbb R$ induces an ordering on $F$. ...
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2answers
28 views

Field $F$ with $\operatorname{char}F=3$ and algebraic over $\mathbb{F}_3$ has a primitive root of unity.

Suppose that $F$ is a field with $\operatorname{char}F=3$ and $F$ is an algebraic extension of $\mathbb{F}_3$. Prove that $F$ contains a primitive $n$th root of unity for some $n>2$.
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The characteristic of real-closed fields is zero?

We know that $F$ is a real-closed field if $F$ is not algebraically closed but $F(\sqrt{-1})$ is algebraically closed. So I have this question What can we say about $\operatorname{char}F$? Is it ...
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1answer
56 views

Why the extension dimension of $x^3-2$ equal to $6$?

I have seen couple questions related to this one, but after reading the answers I am still confused: Why is the extension dimension of $x^3 - 2$ equal to $6$? In other words, why are the basis ...
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2answers
42 views

$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
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1answer
71 views

Finding the Fixed Fields in the Galois Correspondence for the Splitting Field of $x^4-3x^2+4$ over $\mathbb{Q}$

I have found the Galois group of the polynomial $x^4 - 3x^2 + 4$ (see below), but I am not sure how to find the fixed fields in the Galois correspondence. The roots of the polynomial are $$\pm ...
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1answer
39 views

Field extension generated by $\alpha$ and separability

In my notes, I have written that the field extension of $k$ generated by an element $\alpha \in K$, where $K$ a larger field, is defined to be $$k(\alpha) = \bigcap_{ \alpha \in E} E$$ where $E ...
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2answers
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Galois group and degree of splitting field over complex rational functions.

Suppose $F=\mathbb C (t)$ the field of rational functions over $\mathbb C$. let \begin{equation*}f(x)=x^6-t^2\in F[x]\end{equation*}Denote $K$ as the spliting field of $f$ over $F$. I'm trying to ...
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2answers
42 views

Example check: two algebraically closed fields with one a subset of the other

The problem asks to find two different algebraically closed fields $\mathcal{E}$ and $\mathcal{F}$ with $\mathcal{E} \subseteq \mathcal{F}$. We have not done a whole lot of stuff with algebraically ...
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0answers
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Uses for coefficients other than the trace and norm for an element belonging to a field.

The trace and norm are very useful as maps from a field extension to the base field since they are multiplicative/additive and have a lot of other nice properties. They can be defined as two of the ...
3
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1answer
45 views

Given $K(\alpha)/K$ and $K(\beta)/K$ abelian extensions, prove that $K(\alpha + \beta)/K$ is an abelian extension.

Problem: Let $K(\alpha)/K$ and $K(\beta)/K$ algebraic field extensions so that their respective Galois groups are abelian. Prove that the Galois group of the field extension $K(\alpha + ...
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1answer
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“If $\text{Gal}\left(K/F\right)=\left<\sigma\right>$, $N_{K/F}\left(\alpha\right)=1$ , then $\alpha=\frac{\beta}{\sigma\left(\beta\right)}$.”

For (fintie) Galois extension $K/F$, it is easy to show that $N_{K/F}\left(\frac{\beta}{\sigma\left(\beta\right)}\right)=1$ for all $0\ne\beta\in K$, $\sigma\in\text{Gal}\left(K/F\right)$. I want to ...
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45 views

dim. of $\left[\mathbb{Q}\left(\sqrt[n]{a},\zeta_{n}\right):\mathbb{Q}\right]$

I want to find dim. $\left[\mathbb{Q}\left(\sqrt[n]{a},\zeta_{n}\right):\mathbb{Q}\right]$ for positive rational number $a$ and $n^\text{th}$ root of unity $\zeta_{n}$ with assumption ...