Questions tagged [field-theory]

Question 1

In an old japanese book for undergraduate students, I found the following exercice: Let $\mathbb F_q$ be a finite field with $q$ elements and $K=\mathbb F_q\left(\left(\frac1T\right)\right)$ be the ...

Question 2

A problem in a textbook I am reading suggests to check whether $\mathbb{Q}[x]/(x^3+2)$ is a field before formulating the irreducibility criterion (which states that $\mathbb{K}[x]/(f)$ is a field iff $...

Question 3

A problem in a textbook I am reading suggests to check whether $\mathbb{Q}[x]/(x^3+1)$ is a field before formulating the irreducibility criterion (which states that $\mathbb{K}[x]/(f)$ is a field iff $...

Question 4

Do there exist $\alpha$ algebraic over $K$, and $\beta$ algebraic over $K$ such that $K(\alpha) \cap K(\beta) = K$, but min polynomial of $\alpha$ over $K$ is not same as min polynomial of $\alpha$ ...

Question 5

Given an arbitrary rational number $\lambda$, can we find algebraic $\alpha$ and $\beta$ such that $\alpha,\beta\notin\mathbb{Q}$, the field extensions $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$ ...

Question 6

In Milne (Fields and Galois Theory), by definition constructible elements are in $\mathbb{R}$. However, it is also argued (in remark 3.27) that given an algebraic constructible number, its Galois ...

Question 7

I've been wondering about the following question: what are possible fields $K$ such that all algebraic extensions of $K$ are normal? I've found the following examples: Any algebraically closed field. ...

Question 8

Prove that the minimal polynomial of $\frac{\sin{g \theta}}{\sin{\theta}}$ has degree $\frac{p-1}{2}$, where $\theta=\frac{\pi}{p}$, $g$ is a primitive root $\mathrm{mod}$ $p$ and $2 \nmid g$. My ...

Question 9

I began my journey of Algebraic number theory with "Marcus's Number fields ." At page 3 , he defined a relation $\sim$ on the set of ideals of $\mathbb{Z}[w]$ ,where $w=e^{\frac{2\pi i}{p}},...

Question 10

Motivation The polarization formulas are useful because they show that an inner product $(\cdot, \cdot)$ on a vector space $V$ is completely determined by the values along its diagonal. In other words,...

Question 11

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 ...

Question 12

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 "...

Question 13

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 $\lambda,\mu \in \mathbb{C}-\{0\}$. ...

Question 14

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, ...

Question 15

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 $K((y)((x))$. ...

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Questions tagged [field-theory]

Invariant subfield of field of Laurent series

How do I check whether $\mathbb{Q}[x]/(x^3+2)$ is a field without using the irreducibility criterion? [closed]

How do I check whether $\mathbb{Q}[x]/(x^3+1)$ is a field without using the irreducibility criterion? [closed]

Counterexample to a Characterization of Simple Extensions and Minimal Polynomials

Building Field Extensions Around an Arbitrary Rational Number [closed]

Constructibility of conjugates of constructible algebraic numbers

Fields for which all algebraic extensions are normal [duplicate]

Prove that the minimal polynomial of $\frac{\sin{g \frac{\pi}{p}}}{\sin{\frac{\pi}{p}}}$ has degree $\frac{p-1}{2}$ over $\mathbb{Q}$.

An issue with the definition of regular primes in Marcus's Number fields ??

When is a sesquilinear form determined by its diagonal?

Prove that $\mathbb{Q}(\alpha, \beta, \gamma, r_1) = \mathbb{Q}(r_1, r_2, r_3, r_4)$

Proving k(α) is a finite extension. [closed]

$\mathbb{C}(\lambda x^ay^b,\mu x^cy^d)=\mathbb{C}(x,y)$ if and only if $ad-bc= \pm1$

If $K/F$ and $L/F$ are separable, then the composite $KL/F$ is separable

What can be said about $K((x))((y))\otimes_{K((x,y))} K((y))((x))$?

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