1
vote
0answers
27 views

Solving system of equation over distinct finite fields

Is it possible to solve an equation systems which involves elements from distinct finite fields? One example could be elements from $\mathrm{GF}(2)$ and $\mathrm{GF}(2^2)$: With: $x=[1,1,0,1]$ ...
0
votes
0answers
21 views

If $k[a_1,a_2,…,a_n]=k(a_1,a_2,…,a_n)$ show that $a_1,…,a_n$ are algebraic over $k$.

I am trying to prove the following statment and need some help. Let $k$ and $E$ be fields such that $k \subset E$ and $a_1,a_2, \ldots ,a_n \in E$, if $k[a_1,a_2,...,a_n]=k(a_1,a_2,...,a_n)$ show ...
1
vote
1answer
17 views

checking a solution to an exercise in field extension

this is the exercise: suppose $K|F$ is a field extension , $\alpha,\beta\in K^∗$ , $m,n$ are two integers that $(m,n)=1$ and $α^m,β^n∈F$,prove that $αβ$ is a primitive element of $F(α,β)|F$. this is ...
1
vote
1answer
18 views

the number of intermediate fields in a simple field extension of degree $n$

suppose that $K|F$ is a simple field extension with degree $n$,prove that the number of intermediate fields is less or equal $2^{n-1}$. i've done this: assume $K=F(a)$ and $L$ is a intermediate ...
6
votes
0answers
117 views
+200

On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Car}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
1
vote
1answer
28 views

What is wrong with my proof of a step in Artin's construction of algebraic closure?

I'm working through Atiyah & MacDonald, and there's an exercise basically asking you to fill in a certain step in Artin's construction of an algebraic closure for a given field. The question is ...
2
votes
1answer
28 views

Using Kronecker's theorem to construct a field with four elements

Use Kronecker's theorem to construct a field with four elements by adjoining a suitable root of $x^4-x$ to $\mathbb Z/2\mathbb Z$. Definition: A polynomial $f(x)\in F[x]$ splits over $F$ if it is ...
0
votes
1answer
7 views

Tensor product of fields and its subalgebra

In Nathan Jacobson's Basic Algebra II, in section 8.18: Tensor product of fields he is discussing what happens to $E \otimes_FK$, when $K|F$ and $E|F$, and E is algebraic over F. At one point he ...
2
votes
0answers
38 views

Brauer groups of curves and base change

Let $X/k$ be a smooth, projective curve over $k$ and let $L/k$ be a finite extension of fields, where $k$ is a finite extension of $\mathbb{Q}_p$, $p \not=2$. Suppose $k(X)$ contains no elements ...
0
votes
1answer
9 views

Normal closures of transcendental extensions

If $E$ is a finite algebraic extension of the field $F$, then we can find a normal closure of $E$ over $F$. What can we say if the extension $E$ is not finite?
4
votes
1answer
33 views

Fields extensions over isomorphic fields of different degrees

What are the simplest examples of situations where in a field $F$ there are two subfields $L_1$ and $L_2$ such that extensions $F/L_1$ and $F/L_2$ are finite, degrees are different $$ [F:L_1] \neq ...
3
votes
1answer
51 views

Compute the transcendence degree (transcendence degree and tensor products)

$\DeclareMathOperator{\quot}{Quot}\DeclareMathOperator{\tr}{tr}$ Let $I_1$ and $I_2$ be nontrivial ideals in $\mathbb C[x_1,\ldots,x_k]$ and $\mathbb C[y_1,\ldots,y_m]$, respectively. Define $$ R_1 ...
3
votes
0answers
22 views

Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1.
2
votes
0answers
17 views

Is every field between $F$ and $F(\alpha_1,\cdots,\alpha_n)$ of the form $F(\alpha_j,\cdots,\alpha_k)$

Say I have a field $F$, and an extension field $L = F(\alpha_1,\cdots,\alpha_n)$. Is it true that every $K$ such that $$ F \subset K \subset F $$ (all field extensions), $K = ...
1
vote
1answer
28 views

$\mathbb{Q}(\sqrt{m}, \sqrt{n})$ : ring of integers, integral basis and discriminant

In the following document, http://people.math.carleton.ca/~williams/papers/pdf/033.pdf, I found three results about biquadratic fields and their ring of integers. It's the proof of the first theorem ...
0
votes
2answers
28 views

Algebraic Extensions

I have the following question: there is this statement i can't understand: Let $A$ be an integral domain which is integrally closed ( in its field of franctions ) and let $K$ be its fraction field. ...
0
votes
1answer
33 views

How to Prove: If $A$ and $B$ are subfields of a field $F$, then $\{b+a|b\in B, a\in A\}$ is also a subfield of $F$.

I haven't been able to find any counterexamples for either of the two. (1) seemed intuitively true but I had my doubts on (2) and couldn't find one. If there aren't any counterexamples, how can I go ...
0
votes
1answer
20 views

Let K/F be a transcendental extension , then does every F-homomorphism has to be an automorphism?

I have figured out that if $K/F$ be an algebraic extension , then does every F-homomorphism need not be an automorphism . But I can't figure it out for in thee trasncendental case.
1
vote
0answers
40 views

Find $[\mathbb{Q}(\sqrt [3] {3},\sqrt [3] {2}):\mathbb{Q}]$ [duplicate]

I am trying to find $[\mathbb{Q}(\sqrt [3] {3},\sqrt [3] {2}):\mathbb{Q}]$. My guess is it is $9$. There are 3 possibilities-3,6,9. If it is not 9 then $X^{3}-3$ is not irreducible over ...
3
votes
3answers
61 views

$[\mathbb{Q}(\sqrt [3] {2}+\sqrt {5}):\mathbb{Q}]$ [duplicate]

What is $[\mathbb{Q}(\sqrt [3] {2}+\sqrt {5}):\mathbb{Q}]$? A straight forward way will be to just set $x=\sqrt [3] {2}+\sqrt {5}$, take powers and reach at a polynomial in $x$ and show the polynomial ...
5
votes
1answer
82 views

$\mathbf{Q}[\sqrt 5+\sqrt[3] 2]=\mathbf{Q}[\sqrt 5,\sqrt[3] 2]$?

Is there a general or elegant way to approach this problem? One can show that $\sqrt 5+\sqrt[3] 2$ is a root of the hexic $x^6-6x^4-10x^3+12x^2-60x+17$, which should then be its minimal polynomial to ...
0
votes
2answers
40 views

every irreducible polynomial has a root in some field extension

We know the following fact from field theory. Let $F$ be a field and $p(X)$ an irreducible polynomial in $F[X]$. Then we can find a field extension $L$ of $F$ such that $p(X)$ has a root in $L$. ...
2
votes
0answers
12 views

Is the embedding problem with a cyclic kernel always solvable?

This question comes from this question by user72870. I shall explain how it relates to that question at the end. Let me shortly define my question: We call an embedding problem a diagram of the form: ...
0
votes
1answer
35 views

When people say, “K is an extension of k with dimension n”, do they mean as an algebra or as a vectorspace?

For instance, consider k(x), (the fraction field of k[x]). k(x) has dimension 2 as an algebra over k, but dimension \omega as a vectorspace over k. Which one are they talking about, and how can I ...
0
votes
2answers
62 views

Is the map an automorphism?

Please verify the following proof or comment on how would you have proven it. Suppose $q = p^2$ and we have $f: \mathbb{F}_{q} \to \mathbb{F}_{q}$ where $f(a) = a\cdot a^p$ Let $a\cdot a^p = b\cdot ...
-3
votes
2answers
101 views

What are the root of $x^3 - 2$ $\in \mathbb{R}[x]$? [closed]

From the given polynomial it is evident that we have to find a +ve number in $\mathbb{R}$ such that the cube is 2 if it exists. There is one and that is $\sqrt[3]{2}$. How to find the other roots in ...
1
vote
0answers
30 views

Number fields and algebraic integer

Suppose there is a (algebraic) number field $K$. Algebraic integer is a root of some monic polynomial with coefficients in $\mathbb{Z}$. The elements of $K$ are the root of some monic polynomial ...
4
votes
1answer
48 views

Infinite algebraic extension of $\mathbb{Q}$

I have this problem in a exercise list: "Prove that $K=\mathbb{Q}(2^{\frac{1}{2}},2^{\frac{1}{3}}, 2^{\frac{1}{4}}, \ldots)$ is an algebraic extension, but not a finite extension of $\mathbb{Q}$." ...
0
votes
3answers
46 views

clarification of algebraic closure and algebraically closed field

Definition of Algebraic closure: An extension K of F is called an algebraic closure of F if a) F $\subset$ K is algebraic b) K is algebraically closed given the above definition I have been trying ...
3
votes
2answers
42 views

$F(x,y)$ over $F$ is not simple

Let $F$ be a field. Let $x,y$ two algebraically independent indeterminates. Show that $F(x,y)/F$ is not a simple extension. Attempt: I tried by contradiction, assuming that $F(t)=F(x,y)$ and writing ...
2
votes
0answers
53 views

When $\mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta)$?

Let $\alpha, \beta$ be algebraic numbers over $\mathbb{Q}$. Which necessary and sufficient conditions are known such that $$ \mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta) \text{ ?} \tag{$\ast$}$$ ...
2
votes
1answer
66 views

Why is ring of integers $\mathcal O_K$ called ring of integers - what properties of $\mathbb{Z}$ does it inherit?

I was wondering why ring of integers $\mathcal O_K$ for field $K$ is called ring of integers. Definition says that elements in this ring will be a solution for monic equation with coefficients ...
1
vote
1answer
43 views

Algebraic extension of rational numbers.

Let $1<m_1,\ldots,m_r\in{\mathbb{Z}}$. If $K=\mathbb{Q}(\sqrt{m_1},\ldots,\sqrt{m_r})$, and $1<n\in{\mathbb{Z}}$ so that $m_i\nmid{n}$. Is true that $\sqrt{n}\notin{K}$? Added: In addition ...
3
votes
1answer
59 views

Proper Field extensions

Given a field $F$, is there a proper field extension $K$ such that any root in $K$ of a polynomial in $F[X]$ is in $F$? Note: I am not looking for the algebraic closure of $F$. One candidate is the ...
1
vote
1answer
35 views

Field extensions: compute the degree of an extension.

I'm stuck with this problem. Let $F\subseteq E$ and $\gamma\in E$ is trascendental over $F$. Let $m$ be a positive integer. Show that $[F(\gamma):F(\gamma^{m})]=m$, where $[\quad:\quad]$ is the ...
4
votes
1answer
37 views

What is the relationship between the trace/norm of a quaternion and the definition in field theory?

I'm having some trouble figuring out the relationship between the trace/norm of a quaternion element and the definition of trace/norm in the extensions of vector spaces. According to my number theory ...
1
vote
1answer
37 views

Minimal polynomial and field extension

If the degree of a field extension $[\mathbb{Q}(\alpha):\mathbb{Q}]=n\gt 1$ and $\alpha$ is a root of a monic polynomial $f \in \mathbb{Q}[T]$ and the degree of $f$ is $n$. Does the above imply ...
0
votes
0answers
18 views

Degree of the separable closure of the residue field of a complete field

I'm trying to prove a corollary on ramified extensions but I don't know if my thoughts are true. Let $L$ be a finite extension of a complete discrete valuation field $F$, let ...
0
votes
0answers
29 views

Presence of non square elements in $GF(p)$

I came across a problem which had a line of explanation as follows : Let $a \in GF(p). a$ is a non square in $GF(p),~ p \neq 2 \implies \nexists~ b\in GF(p)~~|~~a =b^2 $. But, is it really possible ...
1
vote
1answer
52 views

Counting the roots of a polynomial over a finite field

Let $\mathbb{F}_{11}$ be the field of 11 elements and let $\mathcal{K}$ be the splitting field of $x^{3} - 1$ over $\mathbb{F}_{11}$. How many roots does $(x^{2} - 3)(x^{3} - 3)$ have in ...
1
vote
0answers
24 views

determine the degree of an extension, check normality

I came across an old exam problem and I wonder if my solution is correct. Let $L=\mathbb{Q}(\omega)$, where $\omega=e^{\frac{2\pi i}{6}}$ is a primitive sixth root of unity: a) determine ...
1
vote
2answers
55 views

Proving that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$

Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ and let $a,b \in E$ where $E$ is some extension of $F$. If $a$ is a zero of $f(x)$ and $b$ is a zero of $g(x)$, show that $f(x)$ is ...
3
votes
2answers
175 views

Sum and Product of two transcendental numbers cannot be simultaneously algebraic

If $\alpha$ and $\beta$ are real number and $\alpha$ and $\beta$ are transcendental over $\mathbb Q$, show that $\alpha \beta$ or $\alpha +\beta$ is also transcendental over $\mathbb Q$ Attempt: ...
4
votes
1answer
38 views

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$. Find examples to illustrate that $[F(a):F(a^3)]$ can be $1,2$ or $3$. Attempt: $F \subset F(a^3) \subseteq ...
2
votes
1answer
57 views

A field extension of prime degree

Suppose that $E$ is an extension of $F$ of prime degree. Show that $~~\forall~ a \in E : ~ F(a)=F$ or $F(a)=E$ Attempt: Suppose that $E$ is an extension of a field $F$ of prime degree, $p$. ...
2
votes
1answer
32 views

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\deg f(x)$ and $\deg g(x)$ are relatively prime.

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\gcd(~\deg g(x),\deg f(x)~)=1$. If $a$ is a zero of $f(x)$ in some extension of $F$, show that $g(x)$ is irreducible over $F(a)$ ...
3
votes
1answer
50 views

Question about calculating the degree of a finite field extension.

I have a question about calculating the degree of a finite field extension over $\mathbb{Q}$. This is problem 18 in chapter 1 of Patrick Morandi's Field and Galois theory. The problem asked to show ...
2
votes
0answers
56 views

Question on a finite field extension of $\mathbb{Q}$

I have a polynomial $p(x) \in \mathbb{Q}[x]$ and is irreducible over $\mathbb{Q}$. Let it be of degree $n$ and $\alpha_1, ..., \alpha_n$ be its roots. I know that $$ \mathbb{Q}(\alpha_i) \cong ...
4
votes
2answers
46 views

Non-isomorphic field extensions of $\mathbb{Q}$

I'm having a little bit of a problem with the following question: Show that there do not exist two irreducible polynomials $a(x)$ and $b(x)$ in $\mathbb{Q}[x]$ of degrees 6 and 7 respectively ...
3
votes
3answers
96 views

Finitely many embeddings of a finite extension in an algebraic closure

So I'm reading through Lang's Algebra, and he keeps saying something along the following lines: "Let $K$ be a finite extension of a field $k$ and let $\sigma_1,\ldots,\sigma_r$ be the distinct ...