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. Please use (galois-theory) instead for questions specifically about that ...

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5
votes
2answers
202 views

How does extending a field affect matrix similitude?

Suppose we have fields $\mathbb{K}_1, \mathbb{K}_2$ s.t. $\mathbb{K}_1 \subset \mathbb{K}_2$. On a paper I'm reading there is a flashy reference to some algebraic results concerning similitude of ...
1
vote
1answer
127 views

writing a field as an R module

let $F$ be a field. for which ring $R$, $F$ is an $R$-module. i know already that as an abelian group $F$ is a $\mathbb Z$- module, what else can we say for a general field $F$.
3
votes
1answer
262 views

The degree of the algebraic closure over the separable closure of an imperfect field

Let $K$ be imperfect, $K^a$ its algebraic closure and $K^{\rm sep}$ its separable closure. Show $[K^a \colon K]$ and $[K^a\colon K^{\rm sep}]$ are infinite. Is $[K^{\rm sep}\colon K]$ infinite? ...
1
vote
2answers
81 views

Is $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?

I have a question about the some rings and fields. Is $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?
4
votes
2answers
427 views

What is $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$?

Let $Aut(\mathbb{C}/\mathbb{Q})$ be the field automorphisms of $\mathbb{C}$, and $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$ the subfield of $\mathbb{C}$ fixed by this group. I supsect that it is equal ...
1
vote
1answer
128 views

Polynomial equations with finite field arithmetic

there are given 3 equations (they are connected with cyclic codes): $$s(x)=v(x)+q(x)g(x)$$ $$g(x)h(x)=x^7+1$$ $$s(x)=v(x)h(x)\bmod(x^7+1)$$ I have following data (for $GF(8)$ with generator ...
4
votes
1answer
325 views

There are enough Galois extensions?

There are enough Galois extensions? For me enough means that every finite extension of a certain field is included in a Galois extension of that field, formally: "Given a field $k$, and a finite ...
5
votes
3answers
659 views

Algebraic Closure of Puiseux Series

Using the construction $R_N = K[t^\frac1N]$ $L_N = Quot(R_N)$ and $P = \bigcup_{N\in \mathbb{N}} L_N$ one automatically gets that the puiseux series are a field. Nevertheless they are also an ...
13
votes
1answer
339 views

Why is $\mathbb{C}_p$ isomorphic to $\mathbb{C}$?

I know that two closed fields of caracteristic $0$ and uncountable are isomorphic iff they have the same cardinality. But I don't know why $\mathbb{C}_p$ has the same cardinality as $\mathbb{C}$. Can ...
4
votes
2answers
812 views

Galois Field Fourier Transform

there are two definitions for Reed-Solomon codes, as transmitting points and as BCH code ( http://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction ). On Wikipedia there is written that we ...
0
votes
2answers
90 views

Binomial formula in $GF(2^m)$

there is a binomial formula: $$(x+y)^n=\displaystyle\sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$$ When operations are done in $GF(2^m)$ then all positive integers are reduced $\bmod2$, so binomial formula ...
3
votes
3answers
728 views

Primitive polynomials of finite fields

there are two primitive polynomials which I can use to construct $GF(2^3)=GF(8)$: $p_1(x) = x^3+x+1$ $p_2(x) = x^3+x^2+1$ $GF(8)$ created with $p_1(x)$: 0 1 $\alpha$ $\alpha^2$ $\alpha^3 = ...
0
votes
1answer
80 views

Field extension

there is for example field $GF(2^4)=GF(16)$. Is $GF(16)$ a subfield of itself? Following this definition http://mathworld.wolfram.com/Subfield.html there is nothing written that subfield must contain ...
1
vote
0answers
155 views

Prime ideal splitting in field extension and its normal closure

The question is: Let L / K be a finite (not necessarily Galois) extension of algebraic number fields and N / K the normal closure of L / K. Show that a prime ideal p of K is totally split in L if and ...
2
votes
0answers
603 views

Quick explanation: calculating class group of $\sqrt{-21}$

I'm trying to calculate the class group of $\mathbb{Q} (\sqrt{-21})$, and I've managed to confuse myself. I've used the standard Minkowski bound to show that we only need to consider principal ideals ...
-1
votes
1answer
184 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 ...
3
votes
3answers
1k views

Degree 2 Field extensions

Are all degree 2 field extensions Galois? I know that this is true over the rationals. But is it true in General?
3
votes
1answer
88 views

Can a subquotient field of a field have a higher transcendence degree?

Let $F$ be an algebraically closed field. Let $$K=F(x_1,x_2,...,x_n)$$ be an extension field of transcendence degree $n$. Is it possible to find a sub-$F$-algebra $R\subset K$, together with a ...
2
votes
1answer
415 views

Let $K$ be a fixed field in $\mathbb C$ of an automorphism of $\mathbb C$. Prove that every finite extension of $K$ in $\mathbb C$ is cyclic.

Let $K$ be a fixed field in $\mathbb C$ (complex numbers) of an automorphism of $\mathbb C$. Prove that every finite extension of $K$ in $\mathbb C$ is cyclic. Thank you for your help!
5
votes
1answer
967 views

primitive root of a finite field

This is a problem similar to one of my homework problems, but not on the homework. The problem states that: Find a primitive root $\beta$ of $F_2[x]/(x^4+x^3+x^2+x+1)$. Questions: I know what a ...
16
votes
5answers
659 views

Why isn't the perfect closure separable?

Let $F\subset K$ be an algebraic extension of fields. By taking the separable closure $K_s$, we obtain a tower $F\subset K_s \subset K$ such that $F\subset K_s$ is separable and $K_s\subset K$ is ...
9
votes
1answer
423 views

Perfect closure is perfect

I've been self-studying inseparable extensions and there's something that seems obvious to everybody but not to me. Let's clear out some definitions that are not so universal: Let $K$ be a field ...
2
votes
1answer
295 views

geometric construction of a given angle

Given any angle how can you say that it is constructable or not?
2
votes
1answer
2k views

Showing field extension $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})/\mathbb{Q}$ degree 8 [duplicate]

Possible Duplicate: The square roots of the primes are linearly independent over the field of rationals I am trying to classify the Galois group of the field extension $\mathbb{Q}(\sqrt{2}, ...
2
votes
1answer
477 views

Galois group is isomorphic to the group of invertible affine transformations

Let $p$ be a prime and suppose $f(x)=x^p-a$ is irreducible. Let $AGL(1,\mathbb{Z}_p)$ be the group of invertible affine transformations of $\mathbb{Z}_p$. Show that the Galois group of $f$ over ...
2
votes
0answers
286 views

field embedding

I've come across this problem in Etingof's notes on representation theory (Problem 5.1 on p. 78). It just sounds nice exercise... The question is : Let $f : k(x_1,\ldots,x_n)\rightarrow ...
4
votes
1answer
190 views

About cyclic extensions of $\mathbb{Q}_p$

I'm trying to learn how to apply local class field theory and I thought about trying to enumerate some low degree abelian extensions of $\mathbb{Q}_p$. The easiest case is the quadratic extensions ...
5
votes
2answers
840 views

Finitely generated field extensions

If $F=K(u_1,\ldots,u_n)$ is a finitely generated extension of $K$ and $M$ is an intermediate field, then $M$ is a finitely generated extension of $K$. I'm not exactly sure how to start this ...
1
vote
1answer
145 views

Relationship Between Field Automorphisms and Embeddings

I'm reading some Galois theory in Lang's Algebra, and he often refers to maps acting on elements of a field extension as embeddings in the algebraic closure of the base field (if I'm not mistaken). ...
4
votes
2answers
1k views

Field Norm Surjective for Finite Extensions of $\mathbb{F}_{p^k}$

I'd rather not have the answer, because I feel like this should be a relatively easy question, and I'm just missing some key step, but could anyone give me a hint on showing that the norm (defined as ...
4
votes
1answer
230 views

Degree of Frobenius

Let $k$ be an algebraically closed field of characteristic $p>0$ and $K/k$ be a function field, i.e. $K$ is finite over $k(t)$. Consider the field extension $K \subseteq K^{1/p}$. Why does it have ...
2
votes
1answer
724 views

Finite Cyclic Extensions

Let $\bar{\mathbb{Q}}$ be a (fixed) algebraic closure of $\mathbb{Q}$ and $\tau\in\bar{\mathbb{Q}},\tau\notin\mathbb{Q}.$ Let $E$ be a subfield of $\bar{\mathbb{Q}}$ maximal with respect to the ...
1
vote
4answers
684 views

Roots of Unity in fields

Which roots of unity are contained in the fields: $\mathbb{Q}[i]$, $\mathbb{Q}[\sqrt2]$, $\mathbb{Q}[\sqrt3]$, $\mathbb{Q}[\sqrt5]$, $\mathbb{Q}[\sqrt{-2}]$ and $\mathbb{Q}[\sqrt{-3}]$? I know that ...
2
votes
1answer
252 views

If $[F : F_p] = n$, does $F$ have $p^n$ elements?

If $[F : F_p] = n$, does $F$ have $p^n$ elements? My book seems to be implying that this is true but I'm not sure why. Thanks!
4
votes
1answer
552 views

An algebraic extension of a perfect field is a perfect field

I would like to show that an algebraic extension of a perfect field is a perfect field, using the following result: Given a field $F$ and some family of perfect subfields $\{F_i\}_{i \in I}$ such ...
5
votes
2answers
654 views

Trace as Bilinear form on a field extension

Can anyone help with this: If $L/K$ is a finite field extension, and we have a $K$-bilinear form given by $$(x,y)\mapsto Tr_{L/K}(xy)$$ then the form is either non-degenerate or $Tr_{L/K}(x)=0$ for ...
1
vote
1answer
122 views

Existence of elements in a extension field

Let $F/K$ be an extension field and let $D$ be a subset of $F$ and $z \in K(D)$. Why we can find a subset $\{d_{1},d_{2},...,d_{n}\} \subseteq D$ such that $z \in K(d_{1},d_{2},...,d_{n})$?
1
vote
1answer
84 views

Is there a general method to order an arbitrary field extension?

Here is something I've been wondering about recently. Suppose you have an arbitrary ordered field $F$, and let $F(\sqrt{a})$ be a field extension with $a>0$ in $F$. Is there then some way to order ...
3
votes
1answer
69 views

Why is every Archimedean ordered skew-field necessarily a field?

While browsing around, I read that any ordered skew-field that satisfies the Archimedean property is commutative, but it was offered without proof. Out of curiosity, is there a quick proof or ...
-1
votes
1answer
410 views

If the degree of field extension is a prime number, the extension is simple [closed]

Let $L$ be an extension field of $K$. Suppose that the degree $[L:K]$ is a prime number. How to show that $L$ is a simple extension of $K$?
45
votes
2answers
5k views

The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
0
votes
1answer
157 views

Showing that an intermediate field is not closed

Hungerford defines a field, $E$ as being closed if $E=E''$ where $E'= \{ \sigma \in \mathrm{Aut}(F/K)|\sigma(u)=u \text{ for all } u\in E \} = \mathrm{Aut}(F/E)$ is a subgroup of ...
8
votes
4answers
418 views

Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$

I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 ...
-2
votes
1answer
202 views

Simple extension?

$\mathbb{Q}(\sqrt 6, \sqrt 10, \sqrt 15):\mathbb{Q}=\mathbb{Q}(\sqrt 6+ \sqrt 10+\sqrt 15):\mathbb{Q}$
7
votes
2answers
320 views

Is a completion of an algebraically closed field with respect to a norm also algebraically closed?

Assume we have an algebraically closed field $F$ with a norm (where $F$ is considered as a vector space over itself), so that $F$ is not complete as a normed space. Let $\overline F$ be its completion ...
2
votes
2answers
382 views

field extension problem

I'm suppossed to use an example to show the following statement. If F over K is galois but not algebraic and L is an intermediate field between K and F, then F over L is not galois. Any help at all ...
4
votes
1answer
258 views

Uniqueness of splitting field

When one defines the splitting field for an arbitrary collection of polynomials, how does one show the uniqueness of such a splitting field? (I'm guessing it is still unique.) The induction argument ...
5
votes
2answers
602 views

Is an intersection of two splitting fields a splitting field?

Let $F$ be a field, and let $K_1$, $K_2$ be two splitting fields over $F$ (Suppose they are contained in a larger field $K$). Is $K_1\cap K_2$ necessarily a splitting field over $F$? The statement is ...
3
votes
2answers
917 views

Is the sub-field of algebraic elements of a field extension of $K$ containing roots of polynomials over $K$ algebraically closed?

If I have a field $K$ and an extension $L$ of $K$ such that all (non-constant) polynomials in $K[X]$ have a root in $L$, is the set of algebraic elements of $L$ over $K$ (the sub-field of all the ...
6
votes
3answers
727 views

What's the Difference Between a Vector and an Hypercomplex Number?

What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties. Perhaps this question could be put more generally as: ...