Questions tagged [field-theory]

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 topic. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

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12
votes
3answers
1k 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
141 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
997 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 ...
10
votes
3answers
2k views

Puiseux series over an algebraically closed field

Using the construction $R_n = K[t^\frac1n]$, $L_n = \text{Quot}(R_n)$ 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
396 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 ...
8
votes
3answers
2k 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
125 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 ...
4
votes
3answers
2k 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
97 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 ...
2
votes
0answers
295 views

Prime ideal splitting in field extension and its normal closure [duplicate]

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 ...
3
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0answers
1k 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
218 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 ...
6
votes
3answers
3k 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?
4
votes
1answer
135 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
772 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
3k 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 ...
18
votes
5answers
928 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 ...
12
votes
1answer
1k 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
351 views

geometric construction of a given angle

Given any angle how can you say that it is constructable or not?
6
votes
1answer
5k 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}, \sqrt{...
4
votes
1answer
677 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 $\...
3
votes
0answers
712 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 k(y_1,\...
4
votes
1answer
279 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 i.e....
7
votes
2answers
3k 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 problem. ...
3
votes
1answer
403 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). ...
7
votes
2answers
2k 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
511 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
1k 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
2k 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
339 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
2k 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 ...
10
votes
2answers
2k 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
126 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
102 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
113 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 ...
0
votes
1answer
1k 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 $K$?
127
votes
3answers
15k 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 elementary ...
0
votes
1answer
298 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 $\mathrm{Aut}(F/K)...
12
votes
4answers
2k 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
2answers
4k views

Is $\mathbb{Q}(\sqrt 2,\sqrt 3,\sqrt 5)$ a simple extension?

Is $\mathbb{Q}(\sqrt 2,\sqrt 3,\sqrt 5)$ a simple extension? If yes, what is an explicit generator of this extension?
9
votes
2answers
971 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
469 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 ...
7
votes
1answer
1k views

How to show the uniqueness of splitting fields?

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
3answers
3k views

Minimal Polynomial of $i + \sqrt{2}$ in $\mathbb{Q}$

I am trying to find Minimal Polynomial of $i + \sqrt{2}$ in $\mathbb{Q}$. I was able to determine the minimal polynomial is fourth degree with roots at $i-\sqrt{2}$, $i+\sqrt{2}$,$-i-\sqrt{2}$,$-i+\...
6
votes
2answers
2k 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 ...
4
votes
2answers
3k 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
1k 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: ...
9
votes
3answers
752 views

$\mathbb{Q}(\pi, i\pi)$ over $\mathbb{Q}$

Is $\mathbb Q(\pi,i\pi):\mathbb Q$ a simple extension?
5
votes
2answers
1k views

Set of elements in $K$ that are purely inseparable over $F$ is a subfield

Let $F\subset K$ be an algebraic field extension. Is the set of all elements of $K$ that are purely inseparable over $F$ necessarily a subfield of $K$?
5
votes
3answers
917 views

Irreducibility and Splitting Fields

Show that over any field $F$, the polynomial $x^3-3x+1$ is either irreducible or splits into linear factors. Edited: This is my attempt: Let $f(x)=x^3-3x+1$. Let $a_1,a_2,a_3$ be the roots of $f$. ...