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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1answer
838 views

Proving an algebraic identity using the axioms of field

I am trying to prove (based on the axioms of field) that $$a^3-b^3=(a-b)(a^2+ab+b^2)$$ So, my first thought was to use the distributive law to show that $$(a-b)(a^2+ab+b^2)=(a-b)\cdot a^2+(a-b)\cdot ...
3
votes
3answers
408 views

Field extension and irreducibility

Let $k$ a field, $P \in k[X]$ irreducible of degree $n \geq 2$, $K$ an extension field of $k$ with degree $m$ such as $\gcd(m,n) =1$. How can I show that $P$ stays irreducible over $K$ ? Thank for ...
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2answers
382 views

Question about the definition of a field…

Just out of curiosity - when we define a field, why bother mention multiplication, when its nothing more then repeating the same addition operation? Here's the definition we were taught in calculus ...
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2answers
367 views

Ring theory, field of fractions

Let $R$=$\mathbb{F}$$[[x]]$, where $\mathbb{F}$ is a field. Show that $F(R)$(the field of fractions) may be identified with the ring $\mathbb{F}$$((x))$ of formal Laurent series. A formal Laurent ...
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0answers
95 views

Can we construct a $\mathbb Q$-basis for the Pythagorean closure of $\mathbb Q?$

This is a follow-up question to this one. I asked it there first but moved it here following the advice from Cam McLeman. I tried to prove that $(\mathbb P:\mathbb Q)=\aleph_0$ and I think I ...
14
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3answers
326 views

Is the Pythagorean closure of $\mathbb Q$ equal to the field of constructible numbers?

A Pythagorean field is one in which every sum of two squares is again a square. $\mathbb Q$ is not Pythagorean, which is easy to see. I have read a theorem online which says that every field has a ...
2
votes
2answers
225 views

Frobenius Auto need not be an automorphism if F is infinite

I'm trying to find an example to show the map $\sigma_p : F \rightarrow F$ given by $\sigma_p(a)=a^p$ for $a\in F$ need not be an automorphism in the case that F is infinite. I'm lost as to where to ...
2
votes
1answer
58 views

Residue field of p-adic valuation

Let $\mathbb{Z}_{(p)} = \left\{\frac{a}{b}\in\mathbb{Q}:p\nmid b\right\}$ How is it posible to show that $\mathbb{Z}_{(p)}/p\mathbb{Z}_{(p)}$ is ismorphic to $\mathbb{Z}/p\mathbb{Z}$. I want to show ...
3
votes
3answers
423 views

Is $\mathbb{R}[X]/(P)$ isomorphic to $\mathbb C$ for every irreducible polynomial $P$ of degree $2?$

Trying to solve a problem, I got stuck on the following question. If $P\in\mathbb R[X]$ is an irreducible polynomial and $\operatorname {deg} P=2$, then is it true that ...
1
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0answers
178 views

Separability of compositum of fields

Let $E/F$ be a finite separable extension, and let $K$ be a function field with constant field $F$. Is the compositum $KE$ of $K$ and $E$ a separable extension over $E$?
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1answer
221 views

maximal abelian and unramified extensions

Let $K$ be a local field (e.g. a finite extension of $\mathbb{Q}_p$). Let $K^{ab}$ and $K^{ur}$ denote the maximal abelian and unramified extensions of $K$ inside an algebraic closure $\overline{K}$ ...
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1answer
301 views

Note on Ring Homomorphisms of Matrices Rings

Assume that $\mathbb{F}$ is a field, and let $\mathbb{M}_{t}\left( \mathbb{F}\right) $ be the ring of matrices of order $t$ over $\mathbb{F}$. Does there exist a non-trivial ring homomorphism from ...
3
votes
1answer
249 views

Prove that $\mathbb{Q} \left( \sqrt[n]{p}\right) \neq \mathbb{Q} \left( \sqrt[n]{q}\right)$.

Given two distinct prime numbers $p$ and $q$, how can we prove that $\mathbb{Q} \left( \sqrt[n]{p}\right) \neq \mathbb{Q} \left( \sqrt[n]{q}\right)$ where $\sqrt[n]{p}$,$\sqrt[n]{q}\in \mathbb{R}$ and ...
2
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0answers
87 views

Intersections of finitely generated field extensions are finite?

I was reading the following post at MathOverflow: http://mathoverflow.net/questions/21086/when-are-intersections-of-finitely-generated-field-extensions-finitely-generated/21093 I can't comment there, ...
18
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1answer
1k views

Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.

I have read the following theorem. If $p_1,p_2,\dots,p_n$ are distinct prime numbers, then$$\left(\mathbb Q\left[\sqrt p_1,\dots,\sqrt p_n\right]:\mathbb Q\right)=2^n.$$ I have tried to prove a ...
8
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1answer
435 views

Can a field be isomorphic to its subfield?

Let $K$ be a field and $K(X)$ be the field of its rational functions. Now let $\phi \in K(X)$ be a rational function such that $K(\phi) \neq K(X)$. Now, since $\phi$ is transcendental over $K$, ...
14
votes
4answers
4k views

Finding the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$.

I have to find the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$. The suggested way of doing it is to prove that $\mathbb Q[\sqrt 2 + \sqrt[3] 2]=\mathbb Q[\sqrt 2,\sqrt[3] 2]$ first. ...
1
vote
1answer
83 views

Roots of $x^4-5x+5$

Suppose $z$ is a complex root of $x^4-5x+5$. What is the extension degree of $\mathbb{Q}(z):\mathbb{Q}$? I suspect it is 4 but I don't have any strategy how to prove it.
10
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1answer
211 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 ...
2
votes
1answer
1k views

Finding a Basis for a Field Extension

I am asked to find the degree and basis for a given field extension $\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{6},\sqrt[3]{24}) $ Now I know that the degree for each vector is $3$, and that the basis will have ...
1
vote
1answer
51 views

Characterization of fields $\mathbb{F}$ with the property: there exists a vector space $V$ such that $V^{\star}$ is isomorphic to $\mathbb{F}[X]$

In this post, Georges Elencwajg says that Erdős-Kaplansky tell us that there does not exist a real vector space whose dual is isomorphic to R[X] ! I would like to know if there is a ...
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vote
1answer
173 views

Minimal polynomial of intermediate extensions under Galois extensions.

Let $K$ be a Galois extension of $F$, and let $a\in K$. Let $n=[K:F]$, $r=[F(a):F]$, and $H=\mathrm{Gal}(K/F(a))$. Let $z_1, z_2,\ldots,z_r$ be left coset representatives of $H$ in $G$. Show that ...
2
votes
2answers
277 views

Elements in finite field extensions

Let $A,K$ be finite fields with $K\supset A$. If $[K:A]=3$, I would like clarification as to why, if $x\in A$ is not a square, then $x$ is not a square in $K$. My notes just mention this fact, but ...
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3answers
156 views

Question on finite extensions of fields

I have been reading about fields and I would like assistance with the following: Let $k$ be an arbitrary field and $f(x)$ be an irreducible polynomial in $k[x]$. Then: Prop: There exists a field $K$ ...
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2answers
364 views

Question about a proof about finite normal extensions

In my book they show that if $K \subset L$ is a finite normal extension, then $L$ is the splitting field for some polynomial $f(X) \in K[X]$. They do so as follows: Suppose $a_1, ... ,a_n$ is a ...
8
votes
1answer
95 views

A field which is not algebraically closed but has no extensions of a fixed degree(s)?

Consider the field $k$ obtained as the union of all finite towers of degree $2$ extensions over the rationals. Then $k$ has no degree $2$ extensions, yet $k$ admits extensions of every other finite ...
11
votes
2answers
1k views

Luroth's Theorem

I have just begun to read Shafarevich's Basic Algebraic Geometry. In the first section of the first chapter, he quotes Luroth's theorem, which states that any subfield of $k(x)$ that is not just $k$ ...
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0answers
389 views

Square-free discriminants and integral bases

Let $K = \mathbb Q(\alpha)$ is a number field. Suppose $\alpha \in O_K$ and let $f \in \mathbb Z[X]$ be its minimal polynomial. Show that if the discriminant of $f$ is a square-free integer, then ...
2
votes
2answers
182 views

Prove that $x$ is algebraic over $k(y)$

Let $k$ be a field and $K$ be its extension field. Suppose $y \in K$ is algebraic over $k(x)$ for some $x \in K$ and $y$ is transcendental over $k$. Then $x$ is algebraic over $k(y)$. I think one ...
4
votes
4answers
394 views

Every rational function of $f \in k(x_1,x_2,\dots,x_n)$ is transcendental over $k$.

Let $k$ be a field and let $x_1,x_2,\dots,x_n$ be indeterminates. How do I show that every non-constant rational function $f \in k(x_1,x_2,\dots,x_n)$ is transcendental over $k$.
2
votes
1answer
606 views

If a normal $K/F$ has no intermediate extensions, then $[K : F]$ is prime

Let $K$ be a finite normal extension of $F$ such that there are no proper intermediate extensions of $K/F$. Show that $[K:F]$ is prime. Give a conterexample if $K$ is not normal over $F$.
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2answers
70 views

Number of embeddings of $\mathbb Q (\alpha)$ into $\mathbb C$ which map $\beta \in \mathbb Q(\alpha)$ to a given conjugate

I'm looking at a proof of the following: Let $K = \mathbb Q(\alpha)$ and $\beta \in K$ with minimal polynomial $g \in \mathbb Q[X]$, where the roots of $g$ are $\beta_1, \ldots , \beta_m$. Then ...
14
votes
3answers
410 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 ...
2
votes
2answers
93 views

Algebraic Field Extension of recursive subfields

Let $p_n$ be the n$^{th}$ prime and define subfields $F_n$ of $\mathbb{R}$ recursively by $F_n$=$F_{n-1}(\sqrt[p_n]{p_n})$. Show that $$[F_n:\mathbb{Q} ]=\prod_{i=1}^n p_i$$ and deduce that an ...
1
vote
1answer
688 views

Field Automorphisms

Quite a general question: For a subfield $K$ of $\mathbb{C}$, how can we prove that every automorphism of $K$ fixes every rational number, and secondly, that the set of such automorphisms forms a ...
3
votes
1answer
940 views

An isomorphism between an extension $K/F$ and a subfield of the ring of $n\times n$ matrices over $F$.

I'm working on a problem from Dummit & Foote's Abstract Algebra and I'm a bit confused about one part of the problem. The problem reads: Let $K$ be an extension of $F$ of degree $n$. ...
4
votes
1answer
793 views

Examples of fields which are not perfect

We know that all finite fields are perfect (fields with char $p$). Also fields with char 0 (infinite fields) are perfect. Then what are the fields that are not perfect?
4
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3answers
330 views

How to prove that the Galois group of a normal extension transitively permutes the factors of an irreducible polynomial?

How to do the following problem? Let $K$ be a normal extension of $F$, and let $f(x)\in F[x]$ be an irreducible polynomial over $F$. Let $g(x)$ and $p(x)$ be monic irreducible factors of $f(x)$ ...
2
votes
1answer
237 views

How many field structures does $\mathbb{R}\times \mathbb{R}$ have?

I know that in $\mathbb{Q}\times \mathbb{Q}$ there are several ways to define a field structure. For example, if $p$ is a prime number, then if we define addition as $(a,b)+(c,d)=(a+b,c+d)$ and ...
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votes
2answers
377 views

Analog to the primitive element theorem for transcendental extensions?

The Primitive Element Theorem states that if $E/F$ is a finite separable field extension, then there exists an element $a$ such that $E=F(a)$. There's a similar result I found, that I don't quite ...
2
votes
2answers
370 views

Computing trace and norm in a number field

Let $K = \mathbb Q(\theta)$, where $\theta$ is a root of the polynomial $f = X^3 - 2X + 6$. Then $f$ is irreducible over $\mathbb Q$, so $[K:\mathbb Q] = 3$. I'm trying to compute $N_{K/\mathbb Q} ...
1
vote
2answers
100 views

Is this element quadratic in this field?

Can someone give me an idea how to prove : a) that the minimal polynomial of $\sqrt[3]{2}\in F$ (where $F$ is the splitting field of $x^3-2$ over $\mathbb{Q} $) over the field ...
2
votes
0answers
78 views

Value groups of an archimedean field and its completion coincide

I am aware that for a non-archimedean field $K$ and its completion $\hat{K}$ with respect to a valuation $v$ (and corresponding absolute value $|\cdot|$), with $v$ extending to valuation $\hat{v}$ on ...
5
votes
2answers
775 views

A slick proof that a field which is finitely generated as a ring is finite

It is a known fact that if $k$ is a field that is finitely generated as a ring, which is the same as having a surjective ring homomorphism $f:\mathbb{Z}[x_1,\dots,x_n]\to k$ for some $n\in ...
3
votes
2answers
81 views

What’s the name of complex-ish extension of $\mathbb{Z}/2\mathbb{Z}$?

In communication theory classes I recall this sort of extension of $\mathbb{Z}/2\mathbb{Z}$ where an imaginary $\alpha$ is defined so that $\alpha^2+\alpha+1 = 0$. Then more such imaginaries must be ...
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vote
3answers
71 views

Find members of this field.

Let $m$ be an integer, $m \geq 2$. and let ${\mathbb Z}_m$ be the set of all positive integers less than $m$, ${\mathbb Z}_m = \{0, 1, ..., m -1\}$. If $a$ and $b$ are in ${\mathbb Z}_m$, let $a + b$ ...
2
votes
2answers
124 views

Why does $\# F(X)=\max(\# F,\# X)$ when $\# X$ is a transcendence basis?

I recently asked a question about the transcendence degree of $\mathbb{C}$ over $\mathbb{Q}$. In a nice answer which I hope to understand better, two facts were given: 1: If $F$ is any infinite field ...
5
votes
3answers
756 views

Why does $\mathbb{C}$ have transcendence degree $\mathfrak{c}$ over $\mathbb{Q}$?

It's pretty well known that $\text{trdeg}(\mathbb{C}/\mathbb{Q})=\mathfrak{c}=|\mathbb{C}|$. As a subset of $\mathbb{C}$, of course the degree cannot be any greater than $\mathfrak{c}$. I'm trying to ...
1
vote
1answer
102 views

Containment of extension fields in a Bourbaki proof: $K(S)\subset K(B)$?

The theorem 3 in the section 14 (V.110) of Bourbaki's Algebra II states that Let $E$ be an extension of a field $K$. All transcendence bases of $E$ over $K$ have the same cardinal. I have a ...
2
votes
1answer
101 views

number of differents vector space structures over the same field $\mathbb{F}$ on an abelian group

My question here raised another one. How many differents vector space structures over a field $\mathbb{F}$ we may have on an abelian group? I know that there are abelian groups that we can not endow ...