Questions tagged [field-theory]
Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. This tag is NOT APPROPRIATE for questions about the fields you encounter in multivariable calculus or physics. Use (vector-fields) for questions on that theme instead.
13,725 questions
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Prove that $\mathbb{Q}(\alpha, \beta, \gamma, r_1) = \mathbb{Q}(r_1, r_2, r_3, r_4)$
Problem Statement: Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$ be an irreducible polynomial over the field of rational numbers $\mathbb{Q}$, where $a, b, c, d \in \mathbb{Q}$. Let $r_1, r_2, r_3, r_4$ be ...
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Proving k(α) is a finite extension. [closed]
Let k ⊆ k(α) be a simple extension, with α transcendental over k. Let E be a
subfield of k(α) properly containing k. Prove that k(α) is a finite extension of E.
This is a question from the book "...
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$\mathbb{C}(\lambda x^ay^b,\mu x^cy^d)=\mathbb{C}(x,y)$ if and only if $ad-bc= \pm1$
Let $a,b,c,d \in \mathbb{N}$ such that $ad-bc= \pm1$.
If I am not wrong, for such $a,b,c,d$ we have:
$\mathbb{C}(\lambda x^ay^b,\mu x^cy^d)=\mathbb{C}(x,y)$, where $\lambda,\mu \in \mathbb{C}-\{0\}$.
...
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If $K/F$ and $L/F$ are separable, then the composite $KL/F$ is separable
Prove that, if $K/F$ and $L/F$ are separable, then the composite $KL/F$ is separable.
My attempt: for finite extensions, this useful lemma holds:
Let $K/F$ be a field extension. If $\alpha_1, \dots, ...
- 1,265
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What can be said about $K((x))((y))\otimes_{K((x,y))} K((y))((x))$?
Let $K$ be a field. There is the iterated field of Laurent series
$$
K((x))((y))=\{f:\mathbb{Z}^2\to K:f(x,y)=0\,\text{for}\,y<-N\,\text{or}\,y\ge -N,x<-N_y\},
$$
and similarly $K((y)((x))$. ...
- 2,783
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Splitting field of a polynomial over $\mathbb{Q}_p$?
How to find the degree and the ramification index of the splitting field of a certain polynomial over $\mathbb{Q}_p$? For instance, $f(x)=3+9x+3x^4+x^6$ over $\mathbb{Q}_3$?
If $\gamma$ is one of the ...
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Are all norms of complex fields spherical?
Any complex vector space $\mathbb{C}^{n}$ is isomorphic to a real vector space $\mathbb{R}^{2 n}$. I was wondering, however, if converting complex vector spaces to real ones offers more freedom with ...
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Prove that the extension $\mathbb{Q} \subset \mathbb{Q}(\sqrt{3 + \sqrt{7}})$ is not normal
I want to prove that the extension $\mathbb{Q} \subset \mathbb{Q}(\sqrt{3 + \sqrt{7}}) = L$ is not normal. My strategy is to show that the minimal polynomial $f(x) = x^4 - 6x^2 + 2$ of $\alpha = \sqrt{...
- 1,265
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Which integral domains $R$ are flat over every subring?
It is well-known that if every integral domain containing a given integral domain $R$ is flat over $R$, then $R$ is a Prüfer domain. So I would like to ask the question about the other direction: ...
- 2,783
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Prove $x^{46}+69x+2025$ is irreducible in $\mathbb Z[x]$
I was told to work in $\mathbb F_{23}$, and also show it has a linear factor $\mathbb Z_5$
Write $f(x)=x^{46}+69x+2025$. We begin by supposing that $f=gh$ for some $g,h \in \mathbb Z[x]$. First, $g$ ...
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If $f(x)$ irreducible over $F$ has irreducible factors $g, h$ over a normal extension $K$, there is $\sigma \in \mathrm{Aut}_F(K)$ with $g^\sigma = h$
Let $K|F$ be a normal extension and let $f(x) \in F[x]$ be an irreducible polynomial over $F$. If $g(x)$ and $h(x)$ are monic factors of $f(x)$ in $K[x]$ which are irreducible over $K$, then exhibit ...
- 1,265
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Not all quadratic extensions over $\mathbb{Q}$ are contained in the compositum of all the splitting fields of irreducible cubics in $\mathbb{Q}[X]$
Question: Let $F$ be the composite of all the splitting fields of irreducible cubics over $\mathbb{Q}$. Prove that
$F$ does not contain all quadratic extensions of $\mathbb{Q}$.
(This is exercise 16 ...
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Double Fields: Do infinite ones exist? [duplicate]
Define a double field to be a set $D$ with three binary operations, $+,\times,\Delta$, all commutative, with identity elements $0,1,\omega$ respectively, such that
$$(D,+,\times)$$ is a field and $$(...
4
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Are these sequences multiplicative?
OEIS sequences A352550 to A352560 are of the form "$a(n)=$ number of modules with $n$ elements over the ring of integers in the real quadratic field of discriminant $d$", and A352561 to ...
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Why can $\frac{\partial }{\partial z}$ and $\frac{\partial }{\partial \bar{z}}$ come from the eigenspace of the almost complex structure?
I’m stuck in understanding where the Wirtinger derivatives $\frac{\partial }{\partial z}$ and $\frac{\partial }{\partial \bar{z}}$ come from. Here're the 2 approaches:
Approach 1 (Natural): This one ...
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