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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0
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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
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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
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3answers
460 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
330 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
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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
443 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
379 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
313 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
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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
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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 ...
18
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
689 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
576 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
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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
136 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 $$|\textrm{Aut}(E/F)|\leq [E:F]$$ ...
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0answers
109 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
112 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
634 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
195 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
450 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
625 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 ...
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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
701 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
346 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
243 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
432 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
110 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
263 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
345 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
369 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 ...
7
votes
1answer
218 views

Calculating Separable Closures

In my study of fields, the notion of the separability of an algebraic field extension is one of the more slippery concepts I have encountered thusfar. What is particularly vexing to me is the notion ...
3
votes
2answers
258 views

Composition Fields and Separability

I am trying to prove the following statement: Let $l \subseteq E \subseteq L$ and $l \subseteq F \subseteq L$ be a towers of field extensions, and suppose $E/l$ is a separable extension. $EF/F$ ...
6
votes
3answers
2k views

Isomorphisms: preserve structure, operation, or order?

Everyone always says that isomorphisms preserve structure... but given the (multiple) definitions of isomorphism, I fail to see how the definitions equate with the intuitive meaning, which is that two ...
3
votes
1answer
191 views

If $V$ is an irreducible group representation over a non-algebraically closed field $F$, what's $\dim_F\, (\operatorname{Hom}_G(V,V))$?

Let $F$ be a field and let $G$ be a group. Let $V$ be an irreducible $F$-linear representation of $G$. If $F$ is algebraically closed, then $\dim_F \,(\operatorname{Hom}_G(V,V)) = 1$ by Schur's ...
2
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0answers
95 views

Given a normal basis $\{g(a)| g\in Gal(L/K)\}$ of $L/K$ finite Galois, can we express $g(a)g'(a)$ relative to this basis?

Let $L/K$ be a finite Galois extension. The normal basis theorem says that there is an element $a\in L$ such that $\{ g(a) | g\in \text{Gal}(L/K)\}$ is a basis of $L$ as a $K$-vector space. Let ...
2
votes
1answer
82 views

Decomposition of Normal Field Extensions

I am trying to prove the following homework problem: Let $K/k$ be a normal field extension and $K_i$ and $K_s$ be intermediate extensions so that $K_i/k$ and $K/K_s$ are purely inseparable and ...
1
vote
1answer
44 views

Showing that $(\mathbb{F}_q[x]/(f_i(x)))^{F_q} = \mathbb{F}_q$

In a proof in my syllabus of a number theory course, they use that $$(\mathbb{F}_q[x]/(f_i(x)))^{F_q} = \mathbb{F}_q$$ where $f_i(x)$ is irreducible, $F_q$ is the frobenius automorphism of ...
7
votes
1answer
935 views

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 ...
8
votes
3answers
2k views

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 ...
0
votes
1answer
134 views

Computing the generators of a real cubic field extension of $\mathbb{Q}$

What's a good example where computing the generators of a real cubic field extension of $\mathbb{Q}$ is nontrivial? I usually see these fields specified in terms of generators, is there a good ...