1
vote
1answer
22 views

Finding irreducible polynomials in a field.

I came across the following problem in Dummit and Foote which states: Find an irreducible polynomial for $e^{2\pi i/9}$ and $e^{2\pi i/10}$ over $\mathbb Q(e^{2\pi i/3})$ and $\mathbb Q$. So we ...
0
votes
1answer
18 views

Minimal polynomial of odd degree

I'm stuck on trying to prove this: Let $K\supset F$ and let $u$ be an algebraic element of $K$ with a minimal polynomial of odd degree. Prove that $F(u)=F(u^2)$. I know that in general, ...
1
vote
3answers
81 views

Show that any finite extension of $\mathbb{Q}$ is not algebraically closed.

EDIT: Entirely wrong question. I wanted to ask something else. How do I show that any finite extension of $\mathbb{Q}$ is not algebraically closed. In other words, the algebraic closure of ...
4
votes
1answer
78 views

Equivalence of definitions for “normal extension” and how to lift isomorphisms to them

Briefly: I want to prove that these two definitions for "normal extension" are equivalent: "$K$ is a splitting field for a collection of polynomials in $F[x]$" vs. "Every irreducible polynomial in ...
0
votes
0answers
14 views

Field Extension of Rational Functions

Let $L = F(x)$ be the field of rational functions over a field F. Let $u \in L \backslash F$. Let $K = F(u)$. If u can be written as $\frac{f}{g}$ where $gcd(f,g) = 1$, then prove $[L:K]$ = max {deg ...
0
votes
1answer
64 views

Field of algebraic numbers over $\mathbb{Q}$

Let $F$ be the field of algebraic numbers over $\mathbb{Q}$. I do remember that this means $F$ is a field extension of rationals over $\mathbb{Q}$. How do I show that the field extension of $F$ is ...
1
vote
1answer
22 views

Algebraic closure of a subfield of the field of fraction of a variety

I think that there is the following claim in Basic Algebraic geometry of Shafarevich. Let $X\to Y$ be a dominant morphism of varieties over an alebraically closed field $k$ of $char=0$. Let $\varphi: ...
0
votes
1answer
54 views

Basis for $\mathbb{Q}[\sqrt{8}]$ over $\mathbb{Q}[\sqrt{2}]$

Provided that $x^2-8$ is the minimal polynomial for $\mathbb Q[\sqrt8]$ and $x^2-2$ is minimal for $\mathbb Q[\sqrt 2]$ we should have a basis with four elements. Thus far I know $1$ and $\sqrt 2$ ...
1
vote
3answers
55 views

Dummit and Foote page 526

I'm having trouble with a line of example 2 on page 526. Consider the field $\mathbb{Q}(\sqrt{2},\sqrt{3})$. generated over $\mathbb{Q}$ by $\sqrt{2}$ and $\sqrt{3}$. Since $\sqrt{3}$ is of degree ...
1
vote
1answer
22 views

Binomial formula over an arbitrary field

I'm working on a problem (namely, if $\alpha + \beta$ is algebraic over $F$ then $\alpha$ is algebraic over $F[\beta]$), and the binomial formula appeared. For the problem, I used the fact that, for ...
0
votes
0answers
25 views

Equal fields and dimension

Let $K \subseteq L$ be fields. Show that K=L if and only if $dim_k L=1$ I know that I will have to show both directions of the implication. I'm very new to the topic of fields so I'm still ...
5
votes
2answers
64 views

Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...
0
votes
0answers
14 views

Field Extensions Identities

I'm working on proving some identities but I need some help clarifying the notation and what exactly each statement is saying. Prove the following identities. (a) $K(A) = QF (K[A])$ (b) $R[A_1 ...
0
votes
0answers
38 views

Notational Clarification - Abstract Algebra

I'm going through a paper on homomorphic encryption by Smart and Vercauteren entitled "Fully Homomorphic SIMD operations" and had a question about some notation used in the paper. In section 2 of the ...
0
votes
1answer
33 views

Concerning a Cyclic Galois Group

Why is it that: $\forall K \supseteq \mathbb{Q}(\mathbb{i}), G=Gal(K / \mathbb{Q}) = \langle \sigma \rangle \implies \sigma (\mathbb{i}) = - \mathbb{i}$? (Note: I am guessing that $\sigma ≠ ...
1
vote
0answers
32 views

Union of field extensions over Q

I am asked to prove that $L=\bigcup_{n=1}^\infty\mathbb{Q}(\sqrt[n]2)$ is an algebraic field extension over $\mathbb{Q}$. So far I have: Let $\beta\in L$, then by definition of union there exists a ...
3
votes
1answer
47 views

Algorithm to find representation of an element of field extension $\mathbb{Q}(q)$ in the form $\sum a_i q^i$

Let $\mathbb{Q}(q)$ be a field extension of $\mathbb{Q}$, where $q$ is a real root of some monic irreducible polynomial $p(x) \in \mathbb{Z}[x]$ of degree $d=3$. Given $x \in \mathbb{R}$, (or ...
1
vote
2answers
70 views

When does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$?

Let $p$ be a prime integer, and let $q=p^r$ and $q'=p^k$. For which values of $r$ and $k$ does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$? From Artin's Algebra, Chapter 15, problem 7.12 from the ...
2
votes
1answer
68 views

Why $K(u)$ is a field?

Let $F$ be an extension field of $K$ and $u\in F$. How do we know that adjoining an element of F to K, makes $K(u)$ a field? I know that $Q(\sqrt2)=\{a+b\sqrt2|a,b\in Q\}$ is a field, but in the ...
1
vote
2answers
37 views

Analogy between the quaternion ring and extensions of the rationals

I've started studing fields and their extensions. As an exercise I proved that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$ by showing that $B=(1,\sqrt2,\sqrt3,\sqrt6)$ is a base for the extension field ...
2
votes
2answers
80 views

Field extension of a finite field

Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements. I am not sure how to do this ...
0
votes
1answer
33 views

Extension Fields and Quotients

In the Dummit and Foote 3ed chapter on field extensions (ch. 13), it is stated as a theorem (6) that $ F(\alpha) \cong F[x]/(p(x))$ where $\alpha$ is a root of $p(x)$ and goes on to state that any ...
8
votes
2answers
169 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 ...
2
votes
2answers
40 views

Infinite field extension but algebraic subset

Is there a subset $ S \subseteq \mathbb{C}$ so that all $s \in S$ are algebraic over $\mathbb{Q}$ and $\left[\mathbb{Q}[S] \colon \mathbb{Q} \right]=\infty$? $\mathbb{Q}[S]$ is defined as ...
0
votes
0answers
22 views

For which $d\in\mathbb{Z}$, $\mathbb{Q}(\sqrt{d})$ primitive root of unity of order $p>2$ prime

If $p>2$ is a prime number, then I have to find $d\in\mathbb{Z}$ such that we have a primitive root of unity of order $p$. I know that $d<0$ because otherwise, you can never have a root of ...
1
vote
1answer
29 views

Why is every monomorphism of E into $\overline{\mathbb{Q}}$ a $\mathbb Q$-monomorphism of E into $\overline{\mathbb{Q}}$?

I've been trying to solve the problem below, but I'm not even sure how to get started. Any help would be greatly appreciated. I feel like there is a key insight that will solve the problem, but I'm ...
0
votes
1answer
11 views

Tower rule for transcedence degree

$\newcommand{\tr}{\operatorname{tr}}$ Let $k \subset E \subset K$ be extension fields. Show that $$\tr\ \deg (K/k)= \tr\ \deg (K/E) + \tr\ \deg (E/k).$$ Here if we consider $S_1$ and $S_2$ be two ...
0
votes
0answers
47 views

Galois Theory: An automorphism fixes a field if and only if it fixes the set of generators.

Let $F/K$ be a field extension. Let $a_{1},...,a_{n}\in{F}$ and $E:=K(a_{1},...,a_{n})$. Then how do we show $\sigma\in{Aut_{E}F}$ if and only if $\sigma(a_{i})=a_{i}$ for all $i=1,2,...,n$? Any ...
4
votes
1answer
96 views

finite field extension problem

Maybe somebody knows how to proove the following algebraic theorem: $C \subset U$ is a field extension and $N \subset U$ so, that all $x \in N$ are algebraic over $C$ and $C[N]=\left\lbrace ...
2
votes
1answer
49 views

Separable field extensions

Let $k$ be a field and $k(x_1,x_2,...x_n)= k(x)$ a finite separable extension. Let $u_1,u_2,..., u_n$ be algebraically independent over $k$. Let $w= u_1x_1 + u_2 x_2 +\cdots +u_n x_n .$ Let $k_u = ...
0
votes
0answers
28 views

Characterizing $\operatorname{Gal}(\mathbb{Q}(\sqrt{p_1},\dots,\sqrt{p_n})/\mathbb{Q})$ for $p_i$ primes?

For what $n$ is $$ \operatorname{Gal}(\mathbb{Q}(\sqrt{p_1},\dots,\sqrt{p_n})/\mathbb{Q}) $$ known, where the $p_i$ are primes? By Kummer theory, I think that $$ ...
0
votes
2answers
70 views

Prove subset of Field is a subfield

Let $K$ be a field of characteristic different from 2, $F$ an algebraic extension of $K$ and $L$ a subset of $F$ with the following properties: $L$ contains $K$, $L$ is a K-vector space, $\forall v \in ...
0
votes
1answer
31 views

Degree of extensions and their composite

Let $F<E$ and $F<K$ be finite extensions and assume that $EK$ is defined as composite of two fields. I need to show that $[EK:F] \leq [E:F][K:F]$, with equality if $[E:F]$ and $[K:F]$ are ...
0
votes
0answers
62 views

Galois Extensions and conjugate elements

Let $E/F$ be a finite Galois extension and let $a,b \in E$ have degrees $m$ and $n$ respectively. And suppose that $[F(a,b):F]=mn$. I need to show that there exist $\sigma$ in the Galois Group of ...
2
votes
1answer
38 views

Isomorphism of an extension field of a field of finite transcendence degree

The following is the proposition (1.4) of Mumford's book Algebraic Geometry If $\mathbb C$ has infinite transcendence degree over $k$, then every variety has a $k$-generic point. In the proof ...
5
votes
0answers
130 views

How to show that any field extension $K/\mathbb{Q}$ of degree 4 that is not is Galois has a quadratic extension $L$ that is Galois over $\mathbb{Q}$.

$\newcommand{\Q}{\mathbb{Q}}$Let $K/\Q$ be a field extension of degree $4$ that is not Galois. How to show that there exists an extension $L\supseteq K$ such that $[L:K]=2$ and $L/\Q$ is Galois? I ...
0
votes
0answers
22 views

$k(u)=k(t)\Leftrightarrow ht(u)=1$

In my lecture we proved the following statement: $u\in k(t)\backslash k$ show $k(u)=k(t)\Leftrightarrow ht(u)=1$ But I don't understand the proof we did (I'll put little numbers over the parts I ...
3
votes
1answer
24 views

A counterexample about an inequality- Field extensions

Consider $A$ and $B$ two intermediate fields of the field extension F/K. I have already proved that $[AB:K]\leq {[A:K][B:K]\over [A\cap B: K]}$. I would like to find a simple example (for example, ...
1
vote
1answer
68 views

Exercise about field extensions [duplicate]

Consider $a_1,\ldots,a_n\in \mathbb Z$. i) Suppose $a_1,\ldots, a_n$ are pairwise relatively prime. I have to see by induction on n that $[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q]=2^n$ Once ...
2
votes
3answers
56 views

Simple field extension inequality proof

Let $\alpha \in \mathbb{C}$ be algebraic over $\mathbb{Q}$ and $F\subseteq \mathbb{C}$ be a subfield. Prove that $[F(\alpha):F]\leqslant [\mathbb{Q}(\alpha):\mathbb{Q}]$. This looks like a problem ...
2
votes
1answer
47 views

Characterizing Galois field extensions via tensor product

Let $K\subset L$ be a finite field extension of degree $n$. Prove that it is Galois if and only if $L\otimes_{K} L\simeq L^{n}$, as $L$-algebras, considering on $L \otimes_{K}L=L^{n}$ the left ...
0
votes
2answers
37 views

Dimension of $K\subset L(\alpha)$ where $L$ is a field extension of $K$

Suppose $L$ is a field extension of $K$ and $\alpha$ an element in a field extension of $L$. Can we say $[K\colon L(\alpha)]=[K\colon K(\alpha)]$? I tried to prove this, but I couldn't come up with a ...
0
votes
6answers
44 views

Prove field extension is a field

I have a field extension $\mathbb Q (2^{1/3}) = a + b2^{1/3} + c2^{2/3}$ where $a,b,c\in \mathbb Q$. I want an elementary proof it indeed is a field. How to go about proving it contains its inverse ...
0
votes
1answer
33 views

Make Galois extension large enough to let Galois group act trivially on module

Let $K|k$ be a finite Galois extension of number fields, inside a given "maximal" (infinite) Galois extension $k_S$ of $k$. Let $G = Gal(k_S | K)$ denote the Galois group and let $A$ be a ...
5
votes
1answer
130 views

Field extensions of finite degree and primitive elements

Over a field $F$ of characteristic $0$, if every every element of an extension field $K$ has degree less than $n$ over $F$, does this tell us that $K$ is a finite degree extension of $F$? So it would ...
1
vote
2answers
62 views

Number of intermediate fields in non-separable extensions that are also not purely inseparable

If a field extension is separable then we know there are only finitely many intermediate fields. If a field extension is purely inseparable then it is possible for there to be infinitely many ...
2
votes
3answers
246 views

Finding a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$.

I have to find a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$. I determined that $\sqrt{2}+\sqrt{3}$ satisfies the equation $(x^2-5)^2-24$ in $\Bbb{Q}$. Hence, the basis should be ...
1
vote
1answer
43 views

embedding of a finite algebraic extension

In one of my courses we are proving something (so far, not surprising) and using the fact: if $F$ is a finite algebraic field extension of $K$, there is an embedding of $F$ into $K$. Well, doesn't ...
-1
votes
2answers
44 views

What does the idea of splitting mean when used with fields and polynomials?

i want to understand what does field splitting represent,from my book A Course In Galois Theory by D.J.H Garling this term is explained by following sentences Suppose that $K$ is field, that ...
0
votes
0answers
31 views

Proposition about intermediate field extensions

This is a problem from Algebra, Hungerford. Exercise V.5.21. (a) Let $L$ and $M$ be intermediate fields of the extension $K \subset F$, of finite dimension over $K$. Assume that $[LM : K:] = [L : ...