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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Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$

So I want to show that $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ and determine its Galois group. My thoughts are as follows: Define $\alpha := \sqrt{2+\sqrt{2}}$. Then it is ...
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265 views

This tower of fields is being ridiculous

Suppose $K\subseteq F\subseteq L$ as fields. Then it is a fact that $[L:K]=[L:F][F:K]$. No other hypotheses are needed (I'm looking at you, Hungerford V.1.2). Now obviously ...
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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 $\mathbb{F}_p((X))$?
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Need help understanding separable polynomials

I have the following definition of what it means for a polynomial to be separable: Let $E$ be a field and $P \in E[x]$ be irreducible. Then we say $P$ is separable if it has no repeated root in ...
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485 views

Necessarily a field between a field and its algebraic extension

This is an exercise in some textbooks. Let $E$ be an algebraic extension of $F$. Suppose $R$ is ring that contains $F$ and is contained in $E$. Prove that $R$ is a field. The trouble is really with ...
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Find intermediate fields of $\mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i) \, | \, \mathbb{Q}(i)$

This is the problem I am facing: Compute the intermediate fields of the extension $K | \mathbb{Q}(i)$ where $K = \mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i)$ and find the intermediate fields $M$ such ...
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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 ...
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Faulty definition of a field in Curtis' Abstract Linear Algebra?

On pp.2-3 of Curtis, Abstract Linear Algebra, he gives a definition of a field which seems to fail to exclude a pathological example. He says a field is a set k with two operations (a+b) and (ab) such ...
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Splitting field of a separable polynomial is separable

Probably a stupid question, but.. Why is the splitting field of a separable polynomial necessarily separable? Thanks. Follow up question Show that if $F$ is a splitting field over $K$ for $P \in ...
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1answer
158 views

Set of elements of degree $2^n$ over a base field is itself a field

Let $F \subset L$ be two fields, and define $K = \{\alpha \in L\mid [F(\alpha): F] \text{ is a power of 2} \}$. Our problem is to prove that $K$ is a field. Closure under reciprocation is easy ...
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601 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 $K$, ...
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567 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 ...
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913 views

How does $\cos(2\pi/257)$ look like in real radicals?

We know $\cos(2\pi/p)$ for p a Fermat prime can be expressed in real radicals. The case $p=17$ is a root of an 8th deg eqn, but can be also given as a sequence of nested radicals, $$\begin{aligned} ...
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1answer
217 views

Other Euler characteristics?

At the end of V.3.4 in Algebra: Chapter 0, Aluffi describes the construction of a Grothendieck group over the category of finite dimensional $\operatorname{k}$-vector spaces, ...
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404 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 ...
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287 views

Is it possible to do calculus on any field with a topology?

I'll try to make my point clear: when we consider the field of complex numbers $\mathbb{C}$ we can do calculus there because we have properties of a field and in the same time we have a topology to ...
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If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
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563 views

Proof that a finite separable extension has only finite many intermediate fields

Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
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Is $\mathbb{Q}[2^{1/3}]$ a field?

Is $\mathbb{Q}[2^{1/3}]=\{a+b2^{1/3}+c2^{2/3};a,b,c \in \mathbb{Q}\}$ a field? I have checked that $b2^{1/3}$ and $c2^{2/3}$ both have inverses, $\frac{2^{2/3}}{2b}$ and $\frac{2^{1/3}}{2c}$, ...
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Why is the product of all units of a finite field equal to $-1$?

Suppose $F=\{0,a_1,\dots,a_{q-1}\}$ is a finite field with $q=p^n$ elements. I'm curious, why is the product of all elements of $F^\ast$ equal to $-1$? I know that $F^\ast$ is cyclic, say generated by ...
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Infinite Degree Algebraic Field Extensions

In I. Martin Isaacs Algebra: A Graduate Course, Isaacs uses the field of algebraic numbers $$\mathbb{A}=\{\alpha \in \mathbb{C} \; | \; \alpha \; \text{algebraic over} \; \mathbb{Q}\}$$ as an example ...
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If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. [duplicate]

Possible Duplicate: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I need to prove this result, but the only starting point I think of is to ...
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849 views

Showing a homomorphism of a field algebraic over $\mathbb{Q}$ to itself is an isomorphism.

Suppose $F$ is algebraic over $\mathbb{Q}$ and $\varphi : F\to F$ is a homomorphism. Prove $\varphi$ is an isomorphism. Showing injectivity follows from the fact that the only ideals in a field ...
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783 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 ...
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Two finite fields with the same number of elements are isomorphic

Fraleigh(7ed) Theorem33.12. Let $p$ be a prime and let $n\in\mathbb{Z}^+$. If $E$ and $E'$ are fields of order $p^n$, then $E \simeq E'$. Proof in the text: Both $E$ and $E'$ have ...
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920 views

$F[x]/(x^2)\cong F[x]/(x^2 - 1)$ if and only if F has characteristic 2

Artin's Algebra, Chapter 10 problem 5.16 states: Let $F$ be a field. Prove that the rings $F[x]/(x^2)$ and $F[x]/(x^2-1)$ are isomorphic if and only if $F$ has characteristic 2. As a pedantic ...
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Show that an algebraically closed field must be infinite.

Show that an algebraically closed field must be infinite. Answer If F is a finite field with elements $a_1, ... , a_n$ the polynomial $f(X)=1 + \prod_{i=1}^n (X - a_i)$ has no root in F, so F ...
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Polynomial irreducible - maximal ideal

I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal. $I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$ $\mathbf 1)$ Is $I_1$ a maximal ideal in ...
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Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$?

Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$? I mean, $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ can be viewed as: $\mathbb{Q}[\sqrt{2}][\sqrt{3}]$, as polynomials of ...
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218 views

Non-algebraically closed field in which every polynomial of degree $<n$ has a root

My problem is to build, for every prime $p$, a field of characteristic $p$ in which every polynomial of degree $\leq n$ ($n$ a fixed natural number) has a root, but such that the field is not ...
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Is $\sqrt{2}\in\mathbb{Q}(\sqrt[8]{3})$ or not?

My hunch is that $\sqrt{2}\not\in\mathbb{Q}(\sqrt[8]{3})$. For practice, I want to compute the splitting field and its degree of $x^8-3$ over $\mathbb{Q}$. I know the roots are ...
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What is the main difference between a vector space and a field?

In my opinion both are almost same. However there should be some differenes like any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is ...
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1answer
457 views

For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$

Let $x$ be an element not algebraic over $F$, and $K \subset F(x)$ a subfield that strictly contains $F$. Why is $x$ algebraic over $K$? Thanks a lot!
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1answer
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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 ...
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274 views

Embedding of a field extension to another

Can $\mathbb{Q}(\sqrt {-2})$ be embedded into a cyclic extension of degree 4 over $\mathbb{Q}$?
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Reducibility of $P(X^2)$

This question is inspired by a comment discussion in If $K=K^2$ then every automorphism of $\mbox{Aut}_K V$, where $\dim V< \infty$, is the square of some endomorphism.. Let $k$ be a field of ...
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Why is a variety over a non-alg. closed field a hypersurface?

Exercise $3$ on page $8$ of Kunz's Introduction to Commutative Algebra and Algebraic Geometry is as follows: If the field $K$ is not algebraically closed, then any $K$-variety $V \subset A^n(K)$ can ...
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1answer
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Why doesn't this hold for $p=2$?

I have a question about the following lemma: Assume that the characteristic of $F$ is $p$ and $p>2$. Then $(t^m-1)/(t^n-1)$ is a square in $F[t, t^{-1}]$ if and only if $(\exists s \in ...
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1answer
218 views

Complete ordered field is an Archimedean field that cannot be extended to an Archimedean field

As a bonus problem, our professor of real analysis asked us to prove that the real numbers (a complete ordered field) cannot be extended into an Archimedean field, with no definition of what he meant ...
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How to compute the Galois group of $x^5+99x-1$ over $\mathbb{Q}$?

I was stuck trying to compute the Galois group of $x^5 + 99x -1$. The problem asks to compute the Galois group over $\mathbb{F}_2, \mathbb{F}_3, \mathbb{F}_5, \mathbb{F}_{11}$ and $\mathbb{Q}$. I was ...
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Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field

I would like to show that if $R$ is a field, then $R(x)$ is a proper subset of $R((x))$, where $R(x)$ is the ring of rational functions, and $R((x))$ is the ring of formal Laurent series. If $f ...
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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 ...
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570 views

$\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}$?

Is there an easy way to see that $$\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}?$$ I know that $\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})$ is a subfield of ...
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Subgroup generated by $1 - \sqrt{2}$, $2 - \sqrt{3}$, $\sqrt{3} - \sqrt{2}$

For a number field $K$, Dirichlet's unit theorem says that$$(O_K)^\times = \mathbb{Z}^{r - 1} \oplus (\text{a finite cyclic group}),$$where $r$ is the number of all infinite places of $K$. An infinite ...
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What is the overall idea of Galois theory?

I am a third year undergraduate, doing a course on Field and Galois theory. Now, while I seem to understand most of the concepts locally, I do not seem to get the 'Whole picture' of what is happening ...
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Intermediate field, normal closure and Galois group

Let $K/F$ be Galois with $G=Gal(K/F)$ and let $L$ be an intermediate field. Let $N\subseteq K$ be the normal closure of $L/F$. If $H=Gal(K/L)$ show that $Gal(K/N)=\cap_{\sigma\in G}\sigma ...
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1answer
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Help understanding fields.

Hi guys I have a test this tuesday and I am given practice questions to do , and I have trouble understanding fields. Like I know by definition what they are, but applying them is kind of confusing. ...
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407 views

The Noether-Deuring Theorem

I have to solve the following exercise taken from the book "Introduction to Representation Theory" by P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina and S. ...
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Hyperreal field extension

In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension? Is it ...
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1answer
838 views

$K/E$, $E/F$ are separable field extensions $\implies$ $K/F $ is separable

I wish to prove that if $K/E$, $E/F$ are algebraic separable field extensions then $K/F$ is separable. I tried taking $a\in K$ and said that if $a\in E$ it is clear, otherwise I looked at the minimal ...