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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6
votes
1answer
737 views

Complete ordered field

I'm trying to prove that; If any Cauchy sequence is convergent in an ordered field F, every nonempty subset of F that has an upperbound has a sup in F. Let A be a nonempty subset of F that is not a ...
4
votes
1answer
339 views

Automorphisms of the field of complex numbers

Using AC one may prove that there are $2^{\mathfrak{c}}$ field automorphisms of the field $\mathbb{C}$. Certainly, only the identity map is $\mathbb{C}$-linear ($\mathbb{C}$-homogenous) among them but ...
1
vote
2answers
166 views

$K$ finite extension of $F$ s.t. for every 2 subextensions $M_1, M_2$, $M_1\subset M_2$ or $M_2\subset M_1$. Then there's $a\in K$ such that $K=F(a)$

Let K be a finite extension of a field F such that for every two intermediate field $M_1$, $M_2$ we have $M_1\subset M_2$ or $M_2\subset M_1$. I need to show that there is an element $a\in K$ such ...
1
vote
1answer
345 views

For a finite field of characteristic $p$, $p-1$ divides $|F|-1$?

Let $F$ a finite field of characteristic $p$. Show that $p-1$ divides $|F|-1$. (We shall see later that $|F|$ is a power of $p$.) I am able to solve this by first showing $|F|$ is a power of $p$. ...
1
vote
2answers
600 views

field of characteristic $p$ and polynomial over it

Could any one tell me for which prime $p$ the polynomial $x^4 +x+6$ has a root of multiplicity $>1$ over a field of characteristic $p$?
5
votes
2answers
253 views

Transcendental extension that is not simple

Let $K$ be a field and $x, y$ be independent variables. How can I show that $K(x, y)/K$ is not a simple extension?
3
votes
1answer
141 views

A question regarding the algebraic closure of a field

I have slight problems understanding a thing about algebraic closures of fields. It seems to me that any algebraic closure $C$ of a field $K$ is a Galois extension, but I read that this is not true. ...
2
votes
2answers
111 views

Characterization of an element being algebraic over $\mathbb{Q}$.

Let $Aut(\mathbb{C}/\mathbb{Q})$ be the set of field automorphisms of $\mathbb{C}$ over $\mathbb{Q}$ (in short, all field automorphisms of $\mathbb{C}$). Let $x$ be an element of $\mathbb{C}$ such ...
2
votes
2answers
125 views

Multiplicative Selfinverse in Fields

I assume there are only two multiplicative self inverse in each field with characteristice bigger than $2$ (the field is finite but I think it holds in general). In a field $F$ with ...
8
votes
1answer
213 views

Can we always find a primitive element that is a square?

Let $L/\mathbb Q$ be a galois extension. The Primitive element theorem says, that there is an element $\alpha \in L$, so that $L=\mathbb Q(\alpha)$. Can I always find an element $\beta \in L$, so ...
0
votes
1answer
46 views

probability inequality resolution finite field

I'm trying to find out whether my communication protocol should have redundant information padded, in order to help the receiver correct the error (error correction code, ECC) without needing a ...
6
votes
3answers
499 views

Is $\mathbb{Q}[\sqrt2]$ = $\mathbb{Q}[\sqrt2+1]$?

Is $\mathbb{Q}[\sqrt2]$ = $\mathbb{Q}[\sqrt2+1]$? I think so because $$\mathbb{Q}[\sqrt{2}+1] = \{\sum_{i=0}^{n}c_i(\sqrt{2}+1)^i\mid n\in\mathbb{N}, c_i\in\mathbb{Q}\}$$ $$= ...
3
votes
4answers
1k views

Minimal polynomial of $\sqrt2+1$ in $\mathbb{Q}[\sqrt{2}+\sqrt{3}]$

I'm trying to find the minimal polynomial of $\sqrt2+1$ over $\mathbb{Q}[\sqrt{2}+\sqrt{3}]$. The minimal polynomial of $\sqrt2+1$ over $\mathbb{Q}$ is $$ (X-1)^2-2.$$ So I look at $\alpha = \sqrt2 ...
4
votes
3answers
470 views

Field Extensions as $F$ adjoin some element

Let $F$ be a field and $E$ an extension of $F$. Is it always possible to write $E=F(\alpha_1,\alpha_2,\ldots)$? If $E$ is a finite extension then I think it is possible to write ...
2
votes
1answer
332 views

Linearly dependent vectors over finite fields

My problem is as follows: Assume you have a vector space of dimension $(d + 1)$, with values over $GF(q)$. Every vector in this vector space can be regarded as an element of the extension field ...
1
vote
3answers
65 views

Multiplication in the field $F = \mathbb{Z}_2[x]/f(x)$

Let $f(x) = x^6 + x + 1$ and define the field $F = \mathbb{Z}_2[x]/f(x)$ Compute the following in this field: 1. $(x^5 + x + 1)(x^3 + x^2 +1)$ I start by multiplying (in $\mathbb{Z}_2[x]$): ...
1
vote
2answers
81 views

Degree of $K(X^{1/p}, Y^{1/p})$ over $K(X, Y)$ in characteristic $p$

Let $K$ be a field of characteristic $p$ and $L=K(X,Y)$ where $X$ and $Y$ are variables (i.e. $L$ is the field of fractions of the polynomial ring $K[X,Y]$. Let $\alpha,\beta\in\overline L$ such that ...
3
votes
2answers
460 views

Unramified p-adic extension implies Galois

I am looking for a short proof that if $L \supset K$ are finite extensions of the p-adic numbers $\mathbb{Q}_p$, then if $L/K$ is unramified, $L/K$ is Galois. I think the proof is related to somehow ...
2
votes
1answer
384 views

Quick way to check if a polynomial of degree $> 3$ is irreducible?

What's the easiest way to check if a polynomial of degree > 3 is irreducible in $\mathbb{Z}_2[x]$? I want to find out if $x^7+x^6+1$ is irreducible in $\mathbb{Z}_2[x]$. If a quadratic polynomial ...
0
votes
1answer
320 views

Primitive element theorem - why any finite and separable extension is simple

I have it in my lectures notes that the claim: Let $K/F$ be a finite and separable extension then $K$ is a simple extension of $F$ follows immediately from the theorom : Let $K/F$ be a finite ...
0
votes
1answer
115 views

Why is the following map well defined?

Let $H\leq G=\operatorname{Gal}(K/F)$ ($K/F$ is a finite galois extension), why is the following map well defined: $\varphi:G/H\to\Gamma_F(K^H,K)$ defined by $\sigma H\mapsto\sigma|_{K^H}$ ,where ...
1
vote
2answers
41 views

If $K = \mathbb{F}_p(\alpha)$ where $\alpha^n \in \mathbb{F}_p$ and $n$ is the minimal such $n$. Does this imply that $[K : \mathbb{F}_p] = n$?

If $K = \mathbb{F}_p(\alpha)$ where $\alpha^n \in \mathbb{F}_p$ and $n$ is the minimal such $n$. Does this imply that $[K : \mathbb{F}_p] = n$? If not, is there a condition on $\alpha$ where this is ...
3
votes
3answers
161 views

What is Gal($\mathbb{F}_{q^k}/\mathbb{F}_q)$?

I know that if $q=p$ (where $p$ is prime) then Gal($\mathbb{F}_{p^k}/\mathbb{F}_p)$ is cyclic of order $k$. I heard that in general (for $q=p^m$) the galois group is cyclic of the order of the ...
19
votes
4answers
3k views

How to prove that the sum and product of two algebraic numbers is algebraic?

Suppose $E/F$ is a field extension and $\alpha, \beta \in E$ are algebraic over $F$. Then it is not too hard to see that when $\alpha$ is nonzero, $1/\alpha$ is also algebraic. If $a_0 + a_1\alpha + ...
2
votes
4answers
398 views

What does it mean to take the splitting field of $f(x)\in F[x]$ over $K$ where $K/F$ is a field extension

Let $K/F$ be a field extension and let $f(x)\in F[x]$. I know $f(x)$ have a splitting field, i.e. a field $E$ that $f(x)$ splits in ($E/F$ and $f(x)$ doesn't split in any proper subfield of $E$). I ...
7
votes
1answer
694 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 ...
0
votes
2answers
583 views

An example for a homomorphism that is not an automorphism

Let $K/F$ be a field extension, I know that if $K/F$ is a finite extension then a simple argument from linear algebra shows that since every homomorphism of fields from $K$ to $K$ that fixes $F$ is ...
5
votes
4answers
169 views

Why would the author ask if I used the Associative Law to prove + is not equiv. to *?

I just started reading An Introduction to Mathematical Analysis by H.S. Bear and problem 1 goes as follows: Problem 1: Show that + and * are necessarily different operations. That is, for any ...
2
votes
1answer
137 views

Multiplicative formula for order of automorphism group

I am reading a proof of the following proposition from Dummit and Foote: Let $E$ be the splitting field over $F$ of the polynomial $f(x) \in F[x]$. Then $$|\textrm{Aut}(E/F)|\leq [E:F]$$ ...
0
votes
0answers
112 views

Calculating the numer of irreducible polynomials of degree $12 $ over $\mathbb{F}_p$

I am trying to do an exercise that asks me to calculate the numer of irreducible polynomials of degree $12 $ over $\mathbb{F}_p$. The exercise have 3 parts and I have done the first two parts that ...
3
votes
2answers
113 views

Find all monic polynomials $f\left(x\right)\in F\left[x\right]$ with distinct roots closed under multiplication.

Suppose $F$ is an algebraically closed field. Find all monic polynomials $f\left(x\right)\in F\left[x\right]$ with distinct roots such that the set of roots of $f$ is closed under multiplication. ...
5
votes
4answers
638 views

Minimal polynomial of the root of algebraic number

I have obtained the minimal polynomial of $9-4\sqrt{2}$ over $\mathbb{Q}$ by algebraic operations: $$ (x-9)^2-32 = x^2-18x+49.$$ I wonder how to calculate the minimal polynomial of ...
6
votes
2answers
199 views

In an ordered field, must 1 be positive?

In an ordered field, must the multiplicative identity be positive? Or must it be defined as such?
6
votes
1answer
455 views

Galois Groups of Finite Extensions of Fixed Fields

I am trying to prove the following proposition: Let $L$ be an algebraically closed field, $g \in Aut(L)$ and $K=\{x \in L \; | \; g(x)=x\}$. Show that every finite extensions $E/K$ is a cyclic ...
5
votes
3answers
3k views

How many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$?

I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$. I am interested in counting how many such ...
8
votes
3answers
643 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 ...
4
votes
3answers
85 views

For an ideal $I$ of $K[x]$, $K[x]/I$ is finitely generated iff $I$ is nonnull

In a book on rational series, a blunt statement is made to the effect that: For $K$ a field, $I$ an ideal of $K[x]$, $K[x]/I$ is finitely generated iff $I$ is nonnull. The statement elaborates ...
6
votes
5answers
2k views

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 ...
7
votes
2answers
707 views

How to find irreducible polynomials over $\mathbb{Q}(i)$ with prescribed Galois group?

Here is my recent homework question: For each of the following five fields $F$ and five groups $G$, find an irreducible polynomial in $F[x]$ whose Galois group is isomorphic to $G$. If no example ...
7
votes
1answer
350 views

Generating Elements of Galois Group

I am trying to prove the following: Let $K=\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n})$. Show that $K/\mathbb{Q}$ is Galois with Galois group $(\mathbb{Z}/2\mathbb{Z})^n$. I have attached my proof ...
2
votes
1answer
247 views

Analogy between trace pairing on a number field and the dot product.

How is the trace pairing function $(x,y) \mapsto Tr(xy)$ on a number field an analogue of the dot product in euclidean space? (This is a view shared by Keith Conrad and can be found in his notes ...
3
votes
2answers
437 views

For $K$ the splitting field of $x^8+1$ over $\mathbb{Q}$, determine $Gal(K/\mathbb{Q})$.

Let $f(x) = x^8+1$. To determine the Galois group $G$, we first need the splitting field and before that we need to find the zeroes of $f$. So, $\left(re^{i\theta}\right)^8 = 0$ implies $r=1, ...
2
votes
1answer
114 views

The level of a $p$-adic number field

First I define the level of field. The level of a field $\mathbb K$ is the least $n$ such that $−1$ is a sum of $n$ squares in field, and is denoted by $S(\mathbb K)$. I know that the level of ...
3
votes
2answers
96 views

Bijection between p-adic field embeddings in unramified extensions

A set of notes I am reading claims the following: For $L/K$ and $M/K$ extensions of $p$-adic fields, if $L/K$ is unramified then the natural map $\{K$-embeddings $L \hookrightarrow M\} \to ...
4
votes
1answer
266 views

Field extensions, inverse limits, notation and roots of unity

I'm hoping I can get some assistance with a revision problem and also a notational issue I'm not sure about (although it may not be standard). I seem to remember going over this or something similar ...
2
votes
1answer
140 views

Fields modulo $n$-th powers, discrete valuations and roots of unity

I have been doing some revision on local field theory and have gathered up a collection of questions which I have been unable to make much progress with; there will be a few similar queries along with ...
5
votes
4answers
347 views

What is the meaning of “algebraically indistinguishable”

I heard the term couple of times (in Field theory class and book), for example: The different roots of $p(x)=x^3-2$ are "algebraically indistinguishable". I understand the meaning intuitively, but ...
0
votes
1answer
77 views

Why is $\mathbb{F}(x,y)$ not algebraic?

Let $\mathbb{F}$ be a field, why is $\mathbb{F}(x,y)$ not an algebric extension of $\mathbb{F}$? (Is $\mathbb{F}(x)$ an algebraic extension?) *This is listed in my Field theory lecture notes as an ...
2
votes
2answers
372 views

How to describe when a simple extension $F(\alpha)/F$ is Galois in terms of the minimal polynomial of $\alpha$?

I have a question concerning definition in terms of minimal polynomial i.e. if we let $E = F(\alpha)$ be a field extension of $F$ of degree two then how do I describe, in terms of the minimal ...
4
votes
1answer
149 views

Isomorphic group to the multiplicative group of a field of prime characteristic.

This question is a little bit different from the one I made before. Suppose that I have an algebraically closed field of prime characteristic, is it possible to find an epimorphism $K^*\to K^*\times ...