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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.

An extension field $K$ of the field $F$ (denoted $K/F$) is a field such that $F$ is a subfield of $K$.

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Is $\mathbb{R}$ a maximum Dedekind complete field?

Let $\mathbb{R}\subseteq F$ be an ordered Dedekind complete Field (every Dedekind cut of $F$ is already in $F$), does this mean $\mathbb{R}=F$?
maxuel
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finite extension but not algebraic exetension

I was reading about algebraic extensions. I want to know is there any example where finite extension of any field F is not algebraic extension.
kp9393
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An equality in some extension field

Let $L=K(\alpha)$ be a seperable field extension, end write $f \in K[X]$ for the minimal polynomial of the $\alpha$, let $\alpha_1, \cdots \alpha_n$ denote the roots. Prove the following equality: $$ x^r \quad = \quad \sum_{1 \leq k \leq n}…
Koenraad van Duin
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Let 'a' belong to C and be algebraic over Q, suppose F contained in C is a subfield. Show [F(a) : F] <= [Q(a): Q]

I know that since $a$ is algebraic over $Q$, this means that $Q(a)$ is a finite extension of $Q$ so $[Q(a) : Q] \leq n$ so we can definite a basis $\{v_1, ..........., v_n\}$ for $Q(a)$ Im stuck on how to proceed. Any help would be much appreciated…
user144712
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algebraic element over a field

Let $F$ be a subfield of a field $K$, $a$ an element of $K$. Prove that $a$ is algebraic over $F$ iff $F[a]=F(a)$. For the first direction I've tried this: $a$ is algebraic over $F$ $\Rightarrow$ there is a polynomail $f$ over $F$ such that $f(a)=0$…
stuck
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$x,y\in L$ tow algebraic element such that $xy,x+y\in K$. $x,y\in K$?

if $L/K$ be a extension field and $x,y\in L$ tow algebraic element over K such that $xy,x+y\in K$. Can we say that $x,y\in K$?
Made
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Finding a generator of a field extension

Let $\mathbb{Q} \subset K$ denote the splitting field of $P = (X^2-5)(X^3-2)$. I have to find $\alpha \in \mathbb {C} $ such that $K = \mathbb{Q}(\alpha)$. We know that such an $\alpha$ exists by the primitive element theorem. The roots of $P$ in…
Kieran McShane
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Simple algebraic field extensions of prime degree

Let $K(\alpha)/K$ be a simple algebraic field extension of prime degree $p$. Suppose $\beta \in K(\alpha)$ with $\beta\not\in K$ and $\beta\not=\alpha$. What can we say about $\beta$? Is it necessarily a $K$-conjugate of $\alpha$? Thanks for any…
Mary
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Field extension : Can one define the notation $\mathbb{Q}(\sqrt{5},\sqrt{2})$?

Can one define the notation Can one define the notation $\mathbb{Q}(\sqrt{5},\sqrt{2})$? The question is really similar to this one : field extension-notation problem $\mathbb{Q}(\sqrt{2})=\mathbb{Q}[\sqrt{2}]$ I would say it means "the smallest…
Arlon Fredolster
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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, and so we can consider the field of fractions…
Silver Pine
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Why does the set of algebraic elements of $K/F$ contain $F$?

Let $K/F$ be a field extension and $\tilde{F}$ the set of elements of $K$ that are algebraic over $F$, i.e. $$\tilde{F}=\{\alpha \in K \mid \alpha \ \text{algebraic over} \ F \}$$ in my lecture notes for university there is a corollary which states…
Silver Pine
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Construct the splitting field for the polynomial of $x^4-3$ over $\mathbb{Q}$?

I've been given this problem. I think I must proceed in the following way: Given $a=\sqrt[4]{3}$ and $b=\cos\frac{2\pi}{4}+ \sin\frac{2\pi}{4}i$, the roots of $x^4-3$ are $a$, $ab$, $ab^2$, $ab^3$. We have $\mathbb{Q}\subset…
Red Banana
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Subset of separable elements of normal extension is normal

I currently have a field extension $L/K$ that is normal, and I want to prove that its subset of separable elements over $K$ is itself a normal extension of $K$. In particular, I believe I need to prove that: 1: $J$ is a field and specifically a…
frobeniusguy
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Field Extension example

Find the degree extension of $$[Q(\theta, \phi ) :Q]$$ where $$[Q(\theta) :Q]=5$$ and $$[Q(\phi ) :Q]=2$$. This is what I have $$[Q(\theta, \phi ) :Q]\times[Q(\phi ) :Q]= (5-2)(2) = 6$$. Is this right?
user665125
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Prove that this extension is algebraic

Let $u \in \mathbb{K}(X), \ u = \frac{X^3}{X+1}$. Prove that $\mathbb{K}(X)\supset \mathbb{K}(u)$ is an algebraic extension and find $[\mathbb{K}(X):\mathbb{K}(u)]$. My attemps were trying to show that is a finite extension. If I can prove that then…
HFKy
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