1
vote
1answer
20 views

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 ...
1
vote
1answer
66 views

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?
1
vote
1answer
28 views

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) ...
4
votes
1answer
68 views

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 $\zeta=e^{2\pi i/7}.$ ...
23
votes
1answer
254 views

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?
6
votes
1answer
78 views

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. ...
1
vote
0answers
19 views

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? ...
3
votes
1answer
61 views

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 ...
3
votes
1answer
87 views

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 ...
0
votes
1answer
50 views

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 ...
4
votes
1answer
53 views

Lattice in a vector space of dim 2 over a valuated field.

I'm reading "Arbres, amalgames et SL2" of J.P. Serre, and something is not clear to me, but is to him :) Let $k$ be a field, with a discrete valuation $v$, ie a group epimorphism $v:k^\ast \to ...
8
votes
0answers
129 views

Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
2
votes
1answer
138 views

Roots of $x^n - 1$ in an algebraically closed field of prime characteristic

Let $F$ be an algebraically closed field of characteristic $p$ , and let $n$ be a positive integer. Consider $ g := x^n - 1 \in F[x]$ Is it true that $ g$ has distinct roots in $F$ if and only if ...
2
votes
1answer
35 views

number field:How can i prove that ${\Bbb Q}[\sqrt{-3}]$ is a cyclotomic field?

Can you help me with this ''simple'' exercise: Prove that ${\Bbb Q}[\sqrt{-3}]$ is a cyclotomic field.
3
votes
1answer
91 views

Number of solution to $x^2 + y^2 + z^2 = t$ in a finite field

suppose I have a finite field $\mathbb{F}_q$, where $q = p^m$ and $p$ is prime. Let $0 \not = t \in \mathbb{F}_q$. I was wondering if someone could tell me what the number of solution to $$ x^2 + y^2 ...
2
votes
0answers
60 views

Relationship between the minimal polynomial of $\sin{2^{\circ}}$ and $\sin{5^{\circ}}$ over $\mathbb Q$

Let $f(x)$ be the minimal polynomial of $\sin{2^{\circ}}$ over $\mathbb Q$, and $g(x)$ be the minimal polynomial of $\sin{5^{\circ}}$ over $\mathbb Q$, then $f(x)+f(-x)=2 g(x)\tag 1$. I find this ...
1
vote
2answers
87 views

Applications of additive version of Hilbert's theorem 90

Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$, and $a$ is element of $L$ with trace 0, there exists an ...
1
vote
2answers
54 views

Suppose $γ$ is a $k$th root of unity that satisfies a quadratic equation $z^2−mz−n=0$ with $m,n\in\mathbb{Z}$. Then $k=3,4$ or $6$

Let $k\in\mathbb{Z}$ with $k>2$ and suppose $\gamma$ is a $k$th root of unity that satisfies a quadratic equation $z^2-mz-n=0$ with $m,n\in\mathbb{Z}$.Then $k=3,4$ or $6$. My knowledge on ...
2
votes
1answer
47 views

Definition of $\mathbb Q^c_p$

Let $p$ be a prime number and let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q $ in $\mathbb C$, i.e. the field of algebraic numbers. Is it possible at all to define the $p$-adic completion ...
0
votes
2answers
80 views

Intersection of splitting fields of two polynomials

Let $f,g\in \mathbb{Z}[x]$, $(f,g)=h\in \mathbb{Z}[x]$ (if this is not true, can we construct $h$ in another way?). Let $K_1,K_2$ be the splitting fields of $f,g$ over $\mathbb{Q}$ respectively. Is ...
8
votes
2answers
89 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
4
votes
1answer
108 views

What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been ...
1
vote
2answers
53 views

field number theory question

If we have ${a+b\sqrt{-1}}$ for a,b in ${Z_p}$, with $p$ as an odd prime, with $\sqrt{-1}^2=-1$, how do we show that $a+b\sqrt{-1}$ has a multiplicative inverse iff $a-b\sqrt{-1}$ has a multiplicative ...
2
votes
1answer
132 views

Show that $ \sum q^{-\deg \ p(x)} $ diverges

Show that $\sum q^{-\deg \ p(x)}$ diverges, where the sum is over all monic irreducibles $p(x)$ in $K\left[x\right]$, where $k$ is finite field with $q$ elements. First show that $\sum q^{-\deg ...
1
vote
0answers
78 views

A Gauss sum over a field.

Let $K$ be a field (not necessarily $\mathbb C$) and let $\zeta=\zeta_n$ be a primitive $n$th root of unity in $\bar K$. I would like to know if there is a formula calculating $$ \sum_{k=1}^n ...
5
votes
1answer
97 views

Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
13
votes
2answers
288 views

Sum of irrational numbers, a basic algebra problem

Let $x_1,\dots,x_n$ be positive rational numbers. If $\sqrt[l_1]{x_1},\dots,\sqrt[l_n]{x_n}$ are all irrational numbers (where $l_1,l_2,\dotsc,l_n\in\Bbb N^*$), does it follow that $$\sqrt[l_1]{x_1}+ ...
5
votes
0answers
58 views

Matrix representation of field automorphism

Let $K$ be the degree $n$ field extension of the field $k$, and let $\alpha_1,\dots,\alpha_n\in K$ be a basis of $K$ over $k$ (as a vector space). I read somewhere that the following matrix $$ M= ...
1
vote
1answer
116 views

What is the difference between multiplicative group of integers modulo n and a Galois Field

What is the difference between multiplicative group of integers modulo n and a Galois Field? Is $\mathbb{Z}^*_p$ with p prime the same as $GF(p)$? Or is it the same as $\mathbb{Z}/n\mathbb{Z}$? ...
5
votes
1answer
52 views

Image of the Norm on a Finite Dimensional Extension of $\mathbb{Q}_p$

I've been trying to see whether following assertion is true in order to give a quick proof of another problem I was doing: if $K$ is a finite dimensional extension of the $p$-adic numbers ...
1
vote
1answer
70 views

Solve equations in a field with characteristic p.

Let $p$ be a prime and $K$ a field with characteristic $p$. How to solve the equation $x^2+2=0$ in the field $K$? Here $x, 2, 0 \in K$. Is it equivalent to solve the equation $x^2+2 = 0 \pmod p$? What ...
3
votes
1answer
101 views

Why don't I end up with the same splitting field?

I've understood that the splitting field of $x^4+2$ and the splitting field of $x^4-2$ over $\mathbb{Q}$ are both the field $\mathbb{Q}(\sqrt[4]{2} , i)$. With degree $8$ over $\mathbb{Q}$. This ...
0
votes
1answer
105 views

Sum of the Hyperharmonic\Over-harmonic Series under $\mathbb{Z}_n$ for $p=2$

For $n \geq 5$ prime number, calculate the sum of: $$1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{(n-1)^2}$$ under $\mathbb{Z}_n$. I figured it's the hyperharmonic\over-harmonic series, $$ ...
10
votes
3answers
196 views

Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?

We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
2
votes
2answers
150 views

Roots of unity in $\mathbb{Q}(\zeta_p)$ for $p$ an odd prime

I am confused about a last step in a proof that the only roots of unity in $\mathbb{Q}(\zeta_p)$ are $\pm\zeta_p^j$, where $\zeta_p=e^{2 \pi i/p}$, $p$ is an odd prime and $1\leq j\leq p-1$. So far ...
0
votes
2answers
61 views

The sum of the first and last $k$-th roots of unity

Let $\zeta_k$ be the first complex primitive $k$-th root of unity, i.e. $\zeta_k = e^{2\pi i/k}$. Then the last complex primitive $k$-th root of unity is given by $\zeta_k^{k-1} = \zeta_k^{-1} = ...
9
votes
2answers
390 views

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 ...
1
vote
0answers
89 views

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 $R$ ...
1
vote
1answer
84 views

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 ...
5
votes
1answer
193 views

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 ...
1
vote
2answers
143 views

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 ...
3
votes
1answer
72 views

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 ...
3
votes
2answers
83 views

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 ...
1
vote
1answer
77 views

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 ...
2
votes
4answers
918 views

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 ...
4
votes
2answers
201 views

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 ...
3
votes
1answer
134 views

How would I show that $\sqrt{p}$ is not contained in $Q(\zeta_8)$?

where zeta_8 is the 8th root of unity over Q and p is an odd prime integer.
1
vote
2answers
146 views

The transcendence degree of a separably generated field extension K/L

What is a good reference for the following statement: The transcendence degree of a separably generated field extension $K/L$ is equal to the dimension of the $K$-vector space of derivations $D:K\to ...
3
votes
1answer
240 views

“Place” vs. “Prime” in a number field.

I have been trying to make sense of what a "place" is. In the setting of a number field, is a place simply a prime ideal? My understanding is that one can complete a number field with respect to a ...
2
votes
1answer
156 views

Field Extension of the Rationals

So I'm considering a Field $\mathbb{F}$, such that $\mathbb{Q}$ is a subset of $\mathbb{F}$ and when it's considered a vector space over $\mathbb{Q}$, it has dimension 2. I want to show two things: ...