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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16
votes
5answers
644 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
393 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
289 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
464 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
277 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
187 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
758 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
144 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
212 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
626 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
661 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
237 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
481 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
592 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
82 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
64 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
292 views

$[L:K]$ is prime [closed]

If $[L:K]$ is prime, show that $L$ is a simple extension of $K$.
41
votes
2answers
4k 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
155 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
3answers
385 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
197 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
292 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
370 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
237 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
580 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
852 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 ...
5
votes
3answers
655 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: ...
8
votes
3answers
526 views

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

Is $\mathbb Q(\pi,i\pi):\mathbb Q$ a simple extension?
3
votes
2answers
490 views

Purely inseparable extension

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
682 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 ...
2
votes
0answers
178 views

Is there a subfield F of the complex field with [C:F]=3? [duplicate]

Possible Duplicate: What is the condition for a field to make the degree of its algebraic closure over it infinite? More generally, what is known about subfields $\mathbb{F}\subset ...
4
votes
2answers
364 views

Interesting property of a polynomial ring with coefficients algebraic over $\mathbb{Q}$

Minutes ago I read that if $F$ is the field of algebraic numbers over $\mathbb{Q}$, then every polynomial in $F[x]$ splits over $F$. That's awesome! Nevertheless, I don't fully understand why it is ...
8
votes
2answers
254 views

If $a,b\in\mathbb{Z}$, and if $a+b\sqrt{2}$ has a root in $\mathbb{Q}(\sqrt{2})$, then the root is actually in $\mathbb{Z}[\sqrt{2}]$

I'm working my way though a classical geometry book by Hartshorne right now, but this problem popped up in a section I'm reading. It is Problem 13.10 from Hartshorne's Geometry: Euclid and Beyond if ...
9
votes
2answers
311 views

algebraic version of “finite covering of a compact space is compact”

The following statement is an exercise in point set topology: If $E \to X$ is a covering with nonempty finite fibers and $X$ is compact, then also $E$ is compact. Now Grothendieck generalized covering ...
1
vote
1answer
120 views

Does $\forall n,d\!\in\!\mathbb{N}$ $\forall$ field $\mathbb{F}$ exist an irreducible $f\!\in\!\mathbb{F}[x_1,\ldots,x_n]$ of degree $d$?

how can one show (hopefully in an elementary manner) that there exist irreducible polynomials of arbitrary degree and number of variables over arbitrary field? thank you P.S. induction? EDIT: ehm, ...
6
votes
2answers
997 views

Splitting field of $x^{n}-1$ over $\mathbb{Q}$

From I.N.Herstein's Topics in Algebra. Chap 5 Sec 5.3 Page 227 Problem 8 Problem 8: If $n>1$ prove that the splitting field of $x^{n}-1$ over the field of rational numbers is of degree $\Phi(n)$ ...
4
votes
2answers
283 views

What is the condition for a field to make the degree of its algebraic closure over it infinite?

As we all know, the algebraic closure often has an infinite degree. Also, this shows the necessary and sufficient condition for a Galois extension to be a finite extension of fields. However, we may ...
2
votes
1answer
1k views

“Surjectivity is stable under base change” and field compositums

If $f:X\rightarrow Y$ is a surjective morphism of schemes and $g:X'\rightarrow Y$ is another morphism of schemes, one can show that $p_{2}:X\times_{Y}X'\rightarrow X'$ is also surjective. ...
7
votes
2answers
753 views

A question regarding normal field extensions and Galois groups

The following is possibly true but I can't find a corresponding theorem: If $E/F$ is the splitting field of some polynomial in $F$ and $F \subset K \subset E$ then: $Gal(E/K)$ normal subgroup of ...
3
votes
2answers
218 views

Extending morphisms with Zorn's lemma

I have stumbled upon a remarkable similarity between the proof of Baer's criterion and an extension theorem in field theory. Here are the statements: Baer's criterion: Let $R$ be a ring. A left ...
12
votes
3answers
604 views

A question regarding the definition of Galois group

In my book, Galois group is defined to mean the set of automorphisms on $E/F$ that "leave alone" the elements in $F$. On Wikipedia it says: "If $E/F$ is a Galois extension, then $Aut(E/F)$ is called ...
1
vote
1answer
124 views

Is there a specific name for this notion of extensions of fields?

As this question suggests, I quite like the notion of permuting the coefficients of polynomials. And, moreover, I have another question on this direction:If L|F is a finite normal field extension, ...
1
vote
1answer
351 views

Question regarding separability of polynomials and perfect fields

I have this definition A field $F$ is perfect if it has characteristic $0$ or if it has characteristic $p$ and $F^p = F$ From Wikipedia, I have this fact about separable polynomials: Irreducible ...
3
votes
2answers
266 views

A question regarding finite field extensions

If I understand correctly, the definition of the degree of a field extension $L/K$ is the dimension of $L$ over $K$ interpreted as a vector space. Now if the degree is $n < \infty$, the basis looks ...
2
votes
2answers
591 views

Minimal polynomials

Can someone tell me if this is right: I would like to find the minimal polynomial of (i) $\sqrt[4]{2}i$ over $\mathbb{Q}$ (ii) $\sqrt[4]{2}i$ over $\mathbb{R}$ (i): $\sqrt[4]{2}i$ is a root of ...
1
vote
6answers
232 views

Proving $|K| = p^n \Rightarrow \operatorname{char}(K) = p$

I would like to show that if $K$ is a field with $p^n$ elements then its characteristic has to be $p$, $p$ prime. I'm not sure where to start. It's clear to me that I can construct a field of order ...
0
votes
3answers
568 views

Conjoining elements to fields which are the roots of irreducible polynomials

I know that if $F$ is a field, and $x,y$ are roots of an irreducible polynomial over $F$ lying in an algebraic extension of $F$, then $F(x) \cong F(y)$. But it seems it is not true (in general) that ...