# Tagged Questions

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### Number Systems: Determining when they have closure, identities, inverses, and more.

I have the following $9$ number systems at hand and I am to determine which of them possess a particular property. I am having trouble understanding some of the subtleties between the questions and ...
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### Is there a short proof of the formula for Legendre symbol $(\frac{2}{p})=(-1)^{(p^2-1)/8}$?

Let $p$ > 2 be a prime number. I found in wiki a complex proof for this Legendre symbol: $$\left(\frac{2}{p}\right) = (-1)^{\frac{(p^{2}-1)}{8}}$$ Can anyone give me a short solution please?
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### Some questions about sub-fields of the field of complex numbers

Given a sub-field $f$ of the field $\mathbb{C}$ of complex numbers, is there a name for the smallest sub-field $F(f)$ of $\mathbb{C}$ such that (1) $F(f)$ contains $f$ as a sub-field and (2) ...
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### Milne's Galois Theory Example

The following example is drawn from Milne's Galois Theory notes, p.42 (http://www.jmilne.org/math/CourseNotes/FT.pdf) We study the extension $\mathbb{Q}[\zeta]/\mathbb{Q}$ where $\zeta=e^{2\pi i/7}.$ ...
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### Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
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### Is the extension Galois if $\mathrm{Aut}(K)$ acts transitively on the non-ramified prime ideals?

Let $K/\mathbb Q$ be a finite extension such that $\mathrm{Aut}(K)$ acts transitively on the prime ideals that are not ramified above the same prime $p\in\mathbb N$. Is $K$ Galois? Thanks in advance. ...
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### Discriminant of a splitting field [duplicate]

Let $f(x)\in\mathbf Q[x]$ and let $K$ be the splitting field of $f(x)$. How is the discriminant of $K$ (as a number field) related to the discriminant of $f(x)$? Are they divisible by the same primes? ...
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### Algorithm to find representation of an element of field extension $\mathbb{Q}(q)$ in the form $\sum a_i q^i$

Let $\mathbb{Q}(q)$ be a field extension of $\mathbb{Q}$, where $q$ is a real root of some monic irreducible polynomial $p(x) \in \mathbb{Z}[x]$ of degree $d=3$. Given $x \in \mathbb{R}$, (or ...
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### Find primitive element of splitting field of $1 + x + x^2 - x^5$

As the title says, I need to find the primitive element of the splitting field of $1 + x + x^2 - x^5$ over $\mathbb{Q}$. Firstly, I would proceed by finding the roots as the splitting field has to ...
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### What are $\bar{\mathbb{Q}}$ and $\bar{\mathbb{Q}_{\ell}}$?

Let $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$ and $\bar{\mathbb{Q}_{\ell}}$ the algebraic closure of $\mathbb{Q}_{\ell}$ ($\ell$ is an integer). Could we describe the elements in ...
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### Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?

For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be ...
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### ring of integers in a cubic extension of a cyclotomic function field

Let $K=\mathbb{Q}[\omega]$ where $\omega^2+\omega+1=0$ and let $R$ be the polynomial ring $K[x]$. Let $L$ be the field $K(x)[y]$ where $y$ satisfies $y^3=1+x^2$. Which is the integral closure of $R$ ...
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### Unramification and compositum

The background is: a field $K$ complete with respect to a discrete valuation $|\ |$. We write $A$ and $k$ for his discret valuation ring and the residue field of $A$. We assume that $K$ and $k$ are ...
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### Euclidean norms in quadratic fields

I'm currently reading a set of lecture notes of Number Theory, and there's a small part I'm having trouble understanding. A norm $N: R \rightarrow \mathbb{N}$ is Euclidean if it satisfies: for ...
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### Finite fields are isomorphic

This is from A Course in Arithmetic by JP Serre Theorem 1 ii) Let $p$ be a prime number and let $q = p^f(f \geq 1)$ be a power of $p$. Let be an algebraically closed field ...
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### Generators for $\mathbb F_p^*$

Let $p$ be prime, then it is a well-known fact that $\mathbb F_p^*= \mathbb F_p -\{0\}$ is a cyclic group under multiplication. Are there any methods to determine the generators of this cyclic or any ...
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### Are nonsquares actually squares in extensions of even degree?

I was playing around with finite fields, and noticed that if $F$ is a finite field, and $E$ an extension of odd degree, then every nonsquare $a\in F$ is again a nonsquare in $E$. I found this out by ...
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### Conjecture regarding primal representaion of vectors

I have a conjecture which I cannot prove or disprove. Denote the $i$'th digit of $x$ in binary expansion by $d_i(x)$, where for $i=1$ the MSB is taken. Example: $d_3(110.1)=0$ and $d_4(110.1)=1$ (so ...
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### Difference between fields $\mathbb{Q}[\sqrt{2}+\sqrt{3}]$ and $\mathbb{Q}[\sqrt{2},\sqrt{3}]$? [duplicate]

Possible Duplicate: Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$? How would one describe elements from $\mathbb{Q}[\sqrt{2}+\sqrt{3}]$ and ...
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### find all the $n$ such that $\phi(n) , \phi(n+1) , \phi(n+2)$ are powers of 2

Find all the natural numbers such that, the regular $n , n+1 , n+2$ gons are constructible. Well this problem can be restated in the following way. Since the construction of the regular n-gon is ...