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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7
votes
1answer
327 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 differs from $F$. Why is $x$ algebraic over $K$? Thanks a lot!
2
votes
1answer
101 views

The $p^{th}$ power of the elements of a basis of a finite separable field extension.

I came across the following claim. Let $L/K$ be a finite, separable extension of characteristic $p$ fields. Suppose $a_1,\dots,a_d$ is a basis. Then, so is $a_1^p,\dots,a_d^p$. To prove this, one ...
5
votes
1answer
142 views

Is $i\notin \mathbb{Q}(\zeta_p)$ for all odd primes $p$?

My main question is the title: for an odd prime $p$, denote a primitive $p^{\text{th}}$ root of unity by $\zeta_p$. Is it true that $i$ is not contained in the cyclotomic extension ...
1
vote
2answers
119 views

Unique isomorphism between fields generated by a domain.

Suppose $F$ and $K$ are fields both generated by a common subring $D$, which is a domain. My question is, why is there a unique isomorphism between $F$ and $K$ which is the identity on $D$? Wouldn't ...
1
vote
1answer
141 views

Finding the matrix of multiplication by $\theta^2$, where $\theta^3 - 3\theta + 1 = 0$

This is a problem from a on-line source which yet comes with a solution (self-studier; not h.w.). Let $E = \mathbb Q(\theta)$, where $\theta$ is a root of the irreducible polynomial \[ X^3 -3X + 1. ...
0
votes
1answer
123 views

Intersection of compositum of fields with another field 2

The following is a previous question with an additional hypothesis: Let $F_1$, $F_2$ and $K$ be fields of characteristic $0$ such that $F_1\cap K=F_2 \cap K=M$, the extensions $F_i/(F_i \cap K)$ are ...
1
vote
1answer
110 views

Intersection of compositum of fields with another field

Let $F_1$, $F_2$ and $K$ be fields of characteristic $0$ such that $F_1 \cap K = F_2 \cap K = M$, the extensions $F_i / (F_i \cap K)$ are Galois, and $[F_1 \cap F_2 : M ]$ is finite. Then is $[F_1 F_2 ...
5
votes
2answers
374 views

Primitive element of $\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q}$

Is there a clever way to determine a primitive element of the finite extension $$F=\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q} \text{ ?}$$ On simpler examples, I've been able to find one by ...
25
votes
2answers
1k views

Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
6
votes
1answer
146 views

An “independence” condition on two algebraic elements over $K$.

Let $K$ be a field and let $a,b\in \overline K$ be algebraic elements. I've stumbled upon a certain condition on $a,b$, which I feel could be considered an "independence" condition. I would like to ...
3
votes
2answers
168 views

Dedekind complete ⇒ Sequentially complete

Let F be an ordered field with least upper bound property. 1.Let $\alpha: \mathbb{N} \to F$ be a Cauchy sequence. Since F is an ordered field, $x$ is bounded both above and below. 2.By assumption and ...
2
votes
1answer
109 views

Homomorphism from $\mathbb{Q}$ to an ordered field F

I know that there exists a unique injective function $\gamma : \mathbb Q →F$ for any ordered field F. I don't understand why 'Prove $\gamma(r) = r•1_F$ for every $r\in \mathbb Q$' is an exercise.. ...
6
votes
1answer
554 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
295 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
155 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
283 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
514 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
231 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
128 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
109 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
111 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
212 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
45 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
469 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
971 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
389 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
296 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
60 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
80 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 ...
1
vote
2answers
348 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
302 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
280 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
112 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
156 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 ...
17
votes
4answers
2k 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
370 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
556 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
505 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
164 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
133 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
102 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
111 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
538 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
192 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
383 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 ...
3
votes
3answers
2k 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
525 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 ...
7
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 ...