Tagged Questions

1
vote
1answer
53 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
74 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 ...
1
vote
1answer
33 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, $$ ...
0
votes
0answers
26 views

Proof of Lempel-Golomb construction of Costas array

Can anyone please help me to prove Lempel-Golomb construction of Costas array, i.e., ${g_1}^i + {g_2}^j = 1$ forms costas array where $g_1$ and $g_2$ are primitive roots of a prime $p$ and $1\leq i ...
8
votes
3answers
109 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
101 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
37 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} = ...
4
votes
2answers
128 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
54 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
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1answer
66 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
62 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
97 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
60 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
69 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
74 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
440 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
146 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
117 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
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2answers
82 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
120 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
83 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: ...
7
votes
1answer
179 views

the number field $\mathbb{Q}(\cos \frac \pi n)$

Let $x$ be $\cos \displaystyle \frac \pi n$ for some natural number $n$. Then is it true that $\mathbb{Q}(x^2+x)=\mathbb{Q}(x)$?
2
votes
1answer
57 views

totally real number field generated by square root of an algebraic integer

Let $d$ be a real positive algebraic integer of degree $3$ or $5$. Assume that $\mathbb{Q}(d)$ and $\mathbb{Q}(\sqrt{d})$ are totally real number fields. Is there a possible $d$ which makes that ...
1
vote
1answer
169 views

Proving a polynomial irreducible over finite field [duplicate]

Possible Duplicate: How do I prove that $x^p-x+a$ is irreducible in a field of $p$ elements when $a\neq 0$? How to prove that $x^p - x - 1$ is irreducible over $F_{p}[x]$. I thought ...
2
votes
1answer
119 views

Analogy between trace pairing on a number field and the dot product.

How is the trace pairing function $(x,y) \mapsto Tr(xy)$ on a number field an analogue of the dot product in euclidean space? (This is a view shared by Keith Conrad and can be found in his notes ...
2
votes
1answer
81 views

The level of a $p$-adic number field

First I define the level of field. The level of a field $\mathbb K$ is the least $n$ such that $−1$ is a sum of $n$ squares in field, and is denoted by $S(\mathbb K)$. I know that the level of ...
5
votes
1answer
734 views

Existence of irreducible polynomials over finite field

Let $F$ be a finite field. How do we prove that for each $n \in \mathbb{N}$ there is an irreducible polynomial of degree $n$? One can assume that $F = \mathbb{F}_{p^m}$ where $p$ is prime. If $n \ge ...
1
vote
3answers
264 views

$\mathbb{Q}(\sqrt{d})$ with specific integral basis

I would like some help with the following question. Ireland and Rosen (ch.13#10) For which $d$ does $\mathbb{Q}(\sqrt{d})$ have an integral basis of the form $\alpha, \alpha '$ where $\alpha '$ ...
10
votes
10answers
1k views

Proving $\sqrt 3$ is irrational.

There is a very simple proof by means of divisivility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum: Suppose ...
5
votes
0answers
290 views

Realizing $S_n$ as a Galois group

My question is about the realization of the symmetric group $S_n$ as a galois group of a real and normal field extension $K/\mathbb Q$. As I read, such a field $K$ can be obtained as the splitting ...
8
votes
1answer
144 views

Is $\operatorname{Gal}(\mathbb{Q}_p^{un})\cong \hat{\mathbb{Z}}$?

Is the absolute Galois group of $\mathbb{Q}_p^{un}$ the profinite completion of $\mathbb{Z}$? I was never quite sure... In similar cases, it is true. Namely, $\mathbb{C}((t))$ does have absolute ...
13
votes
3answers
382 views

Is $\bar{\mathbb{Q}}(x)\cap \mathbb{Q}((x))=\mathbb{Q}(x)$? [unsolved (even though we earlier thought it was)]

Fix the algebraic closure of $\mathbb{Q}((x))$ for this question to make sense. I know that $\mathbb{Q}((x)) \cap \overline{\mathbb{Q}(x)}$ has elements that are not in $\mathbb{Q}(x)$ (in analogy to ...
3
votes
1answer
224 views

Is $\mathbb Q_r$ algebraically isomorphic to $\mathbb Q_s$ while r and s denote different primes?

It is obvious that $\mathbb Q_r$ is topologically isomorphic to $\mathbb Q_s$ while $r$ and $s$ denote different primes. But I really don't know whether it is true in the aspect of algebra. As I ...
0
votes
1answer
360 views

An attempt of catching the where-abouts of “ Mysterious group $Ш$ ”

This question is a bit concerned with the Tate-Shaferevich group, lets start defining $C$ as $$C: X^2- \Delta Y^2=4$$ which are generally called as Pell-conics, so all in this question $K$ refers to ...
25
votes
4answers
733 views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
2
votes
2answers
439 views

What is Galois Field

When I am reading some algorithm related to error correction. To generate some polynomial it uses finite-field which is Galois field. I am not from mathematical background. Can anybody explain me in ...
5
votes
1answer
194 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
4
votes
1answer
438 views

Find primitive element such that conductor is relatively prime to an ideal (exercise from Neukirch)

This is an exercise from Neukirch, "Algebraic Number Theory", Ch I, Sec 8, Exercise 2, pg 52. It really has me stumped. Suppose $A$ is a Dedekind domain, $K$ its field of fractions, $L$ a finite, ...
3
votes
1answer
129 views

On a characterization of the tamely ramified coverings of the fraction field of a strict Henselian ring

Let $K$ be the field of fractions of a strictly Henselian discrete valuation ring $A$. Following Milne ("Etale Cohomology", Chapter I, Paragraph 5, example e)), $Spec(K)$ is the algebraic analogue of ...
-1
votes
1answer
165 views

Finite extension of $\mathbb Q_p$

Let $\mathbb K/\mathbb Q_p$ be a finite extension of $p-$adic field $\mathbb Q_p$. Let ${\mathcal O}=\{x\in K\;:\;|x|\leq1\}$ and ${\mathcal P}=\{x\in K:\;|x|<1\}$, here $|\cdot|$ is the absolute ...
4
votes
1answer
259 views

Is every finite separable extension of a strictly henselian DVR totally ramified?

Let $R$ be a discrete valuation ring with field of fraction $K$ and residue field $k$ and let $K'$ be a finite and separable extension of $K$. If $R$ is henselian ("Hensel's lemma holds", e.g. if ...
4
votes
1answer
106 views

Is an unramified cover of the p-adics determined by its degree?

If $K_1$ and $K_2$ are subfields of a pre-chosen $\overline{\mathbb{Q}_p}$, and if they're both unramified at $p$, and $[K_1:\mathbb{Q}_p]=[K_2:\mathbb{Q}_p]$, does that imply that $K_1=K_2$? My ...
8
votes
1answer
193 views

Is $\mathbb{Q}_p(\zeta_p)$ the same as $\mathbb{Q}_p(p^{\frac{1}{p-1}})$?

It seems so. $\mathbb{Q}_p(\zeta_p)$ is a $p-1^{th}$ extension of $\mathbb{Q}_p$ which doesn't extend the residue field; and so is $\mathbb{Q}_p(p^{\frac{1}{p-1}})$. However I can't see how to express ...
2
votes
2answers
173 views

Powers of $x$ as members of Galois Field and their representation as remainders

first question on math.stackexchange :) I'm studying for a Cryptography - Communication Security exam, and it involves a certain quantity of number theory - finite field theory, so be warned: this is ...
5
votes
1answer
255 views

Algebra structure of tensor product of two Galois extensions

Sorry if this question is too basic. It is from Fröhlich and Taylor's "Algebraic Number Theory". Let $E/F$ be a finite Galois extension of fields, with $G=Gal(E/F)$, and let $K$ and $L$ be two ...