0
votes
0answers
10 views

$E/F$ is a finite field extension $\implies$ $\text{Aut}_F E$ is finite?

Question: If $E/F$ is a finite field extension, then is $G :=\text{Aut}_F E$ finite? Attempt: Let $\mathcal{B} := \{x_1, \ldots, x_n\}$ be a basis for $E/F$. Then $\sigma \in G$ is completely ...
0
votes
4answers
29 views

Easy question about tower of fields

If $E/F/G$ is a tower of fields and $[E:G]<\infty$, then does $[F:G]<\infty$? I suspect the answer to be "yes", but somehow the fact that a basis of $E$ over $G$ might have vectors in ...
1
vote
0answers
26 views

Denseness of algebraic and transcendental elements in R [NBHM 2014]

Which of the following statements are true? a. Algebraic numbers over Z are dense in R. b. Transcendental numbers over Z are dense in R. Let me write what I did. For a), the concerned set is a subset ...
0
votes
2answers
30 views

Intersection of any family of subfields is itself a subfield

Prove that the intersection of any family of subfields is itself a subfield. In the countable case: Suppose that $\mathscr K$ is a field and consider $(\mathcal K_n)_{n\in\mathbb N}\subset \mathscr ...
1
vote
3answers
165 views

Can we say “commutative ring = field”?

We know the difference between ring (R) and field (F) is that R does not guarantee multiplication is commutative. Now, if considering commutative R, which means (R,.) is a group, can we say: ...
0
votes
0answers
25 views

Field isomorphism and order of elements

I know that group isomorphism preserves order of element but can someone plese tell me does field isomorphism preserves order of elements?
0
votes
1answer
28 views

Prove $\mathbb{Q}(\sqrt2,\sqrt2i)=\mathbb{Q}(\sqrt2+\sqrt2i)$

I want to know why the following two are equivalent: $$\mathbb{Q}(\sqrt2,\sqrt2i)=\mathbb{Q}(\sqrt2+\sqrt2i)$$, where $\mathbb{Q}$ is the rational number field, and ...
0
votes
1answer
32 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
59 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
94 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
1answer
38 views

Write Galois group as semidirect-product

Consider the polynomial $ x^7-13 \in \Bbb{Q}(x)$. Find Galois group and write it as a semi-direct product. Edit: So here is what I have done. I found the dimension of the splitting field over ...
-1
votes
0answers
28 views

Cubic resolvent of quartic.

Where does the cubic resolvent of quartic come from? I would like to know its derivation since I am having a hard time memorising the formulas.
3
votes
2answers
83 views

Field that contains $(\mathbb{Z}/p^n\mathbb{Z})^*$

Let $p$ be an odd prime. There exists a field $F$ that contains an isomorphic copy of $(\mathbb{Z}/p^n\mathbb{Z})^*$ as a multiplicative subgroup? Clearly if $n=1$ we can take $F=\mathbb{F}_p$, but ...
2
votes
0answers
21 views

Finding degree of an extension

Find the degree of the field extension $\mathbb{Q}[\sqrt[3]{2},\sqrt[3]{3}]$ over $\mathbb{Q}$. My approach: Call the desired degree $n$. Clearly, $3|n$ and $n\leq 9$. So possible values of $n$ are ...
2
votes
0answers
21 views

How do I show that both the additive and multiplicative groups of an infinite field are non-cyclic? [duplicate]

I've tried mimicking the proof in case of $\mathbb Q$ to deal with the characteristic zero case, but can't do the characteristic $p$ case. Can someone give a solution to that end?
2
votes
1answer
39 views

Field Theory : What is wrong with this “homomorphism”?

Let $E/F$ be a field extension and $\alpha \in E$ be algebraic over $F$. Let $m(x) \in F[x]$ be irreducible and such that $m(\alpha) \neq 0$. Define $$ \phi : F(\alpha) \to F[x]/(m) $$ by ...
0
votes
1answer
32 views

Proof Unicity of Splitting Field

Context : Definition : We say $f(x) \in F[x]$ splits over the field extension $E/F$ if $$ f(x) = c (x-\alpha_1)\cdots(x-\alpha_n) $$ for some $c, \alpha_1, \ldots, \alpha_n \in E$. A splitting ...
4
votes
1answer
46 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
41 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 ...
1
vote
0answers
51 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
64 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 ...
0
votes
0answers
23 views

On the Characterization of $F[\alpha]$ for a Field Extension $E/F$.

Let $E/F$ be a field extension. Reading a proof that $$ \alpha \text{ is algebraic over } F \implies [F[\alpha]:F] < \infty, $$ one might begin by considering an element $f(\alpha) \in F[\alpha]$ ...
0
votes
2answers
140 views

Give an example of a field where -1=1

The question is to find a counterexample to the following: In every field $F$, $-1$ is not equal to $1$. My intuition leads me to integers modulo $1$. Is this correct, are the integers modulo $1$ a ...
2
votes
1answer
42 views

Proof of a Field Extensions Theorem

Consider the following result. Theorem : Let $E/F$ be a finite field extension of degree $n$ and let $V$ be a vector space over $E$. Then $$ \dim_F V = [E:F] \dim_E V. $$ Now, it seems like a ...
2
votes
2answers
27 views

A field $K$ is an algebra

I learned this definition of an algebra recently. The definition is: A vectorspace $V$ over a field $K$ is is an algebra if there exists $K$-bilineair map $\varphi\colon V\times V\rightarrow V$ which ...
1
vote
1answer
38 views

If $Q$ is a prime ideal of $R[x]$ then $QF[x]\cap R[x]=Q$

I'm filling the gaps in a proof and I'm stuck in this part: Suppose $R$ is a UFD and $Q$ is a prime ideal of $R[x]$, if $F$ is the quotient field of $R$ and $R\cap Q=\{0\}$, then $QF[x]\cap ...
-1
votes
1answer
76 views

There are no field structures on $\mathbb{R}^3$, but what of $\mathbb{R}^n$ for $n\geq 4$?

Has it been proved that there do not exist nice field structures on $\mathbb{R}^n$ for $n\geq 4$? The quaternions $\mathbb{H}$ fail due to lack of commutativity and the bicomplex numbers ...
3
votes
1answer
54 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 ...
0
votes
1answer
23 views

Separable polynomial and algebraic extension

If $f\in F[t]$ is separable and $E/F$ is an algebraic extension, then how can I be sure that $f$ is separable as an element of $E[t]$? I thought it is a trivial question...but now I think it is ...
8
votes
1answer
327 views

Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
0
votes
2answers
84 views

Fields - Proof that every multiple of zero equals zero

This is one of my first proofs about fields. Please feed back and criticise in every way (including style and details). Let $(F, +, \cdot)$ be a field. Non-trivially, $\textit{associativity}$ implies ...
3
votes
2answers
64 views

In which Fields, does $x^n-x$ have a multiple zero?

In which Fields, does $x^n-x$ have a multiple zero? Attempt: Let $f(x) = x^n-x = x(x^{n-1}-1)$ and $f'(x) = nx^{n-1}-1$ If $f(x)$ has a multiple zero, then, $f(x)$ and $f'(x)$ have a common factor. ...
4
votes
1answer
39 views

Finding a cubic polynomial whose splitting field over $\mathbb{Q}$ equals $\mathbb{Q}(a)$ if $a$ is any of its roots

Question: Let $\alpha$, $\beta$ and $\gamma$ be the roots of a rational cubic polynomial $q$. Can we find a (non-trivial) example where the splitting field of $q$ over $\mathbb{Q}$ equals ...
1
vote
1answer
30 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
35 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
0answers
37 views

Irreducible polynomial

Does there exist an irreducible polynomial over a field K with two roots $a,b$ and $k\in K$ such that $a=b+k$ ? This can't happen if K is of characteristic $0$ , but can it happen if K is of ...
1
vote
1answer
34 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
16 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 ...
2
votes
1answer
38 views

Splitting field of $(x^2-2)(x^6-20)$ over $\mathbb{Q}$

I have to determine the splitting field $K$ of $f(x)=(x^2-2)(x^6-20)$ over $\mathbb{Q}$. My attempt of solution: $K=\mathbb{Q}(\sqrt2, \sqrt[6]{20}, i\sqrt3)$; $d_1:=[\mathbb{Q}(\sqrt2, ...
1
vote
1answer
34 views

Field of prime characteristic over two indeterminates

Let $F$ have prime characteristic $p$ and let $E = F(Y,Z)$, where $Y, Z$ are indeterminates. Let $L=F(Y^{p} , Z^{p})$ $\subseteq E$. a. Show that $\alpha^{p} \in L$ for all $\alpha \in E$. b. Show ...
0
votes
2answers
52 views

What would be a prime element in the field of rational numbers?

It is clear to understand what prime elements will be in case of ring of integers and many other rings. However, I find it confusing in case of fields, more specifically field of rational numbers. So ...
1
vote
1answer
40 views

Proving that either $x $ or $2x$ is a generator of the cyclic group $(\mathbb Z_3[x]/\langle f(x) \rangle)^*$

if $f(x)$ is a cubic irreducible polynomial over $\mathbb Z_3$, prove that either $x $ or $2x$ is a generator of the cyclic group $(\mathbb Z_3[x]/\langle f(x) \rangle)^*$ Attempt: $f(x) = \alpha ...
0
votes
0answers
28 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
0answers
36 views

Showing the existence of sub fields of a finite field

For each divisor $m$ of $n$, $GF(p^n)$ has a unique sub field of order $p^m$ . The proof of this theorem in Gallian goes like this : Suppose that $m$ divides $n$. Then, since : $(p^n-1) = ...
3
votes
3answers
111 views

Is division allowed in rings and fields?

Is division allowed in ring and field? The definition of ring I am using here does not require the presence of multiplicative inverse. I think in general, division is not a well-defined ...
5
votes
2answers
105 views

Generators of the Relations of a Galois Extension

Let $K$ be a Galois extension of $\mathbb{Q}$ of degree $n$. Pick some primitive element and take the roots $a_1, ..., a_n$ of its minimal polynomial. Then the evaluation map $\mathbb{Q}[x_1, ..., ...
5
votes
1answer
194 views

Sigma-Algebra: Is it an Algebra, Field, or Something Else?

The Wikipedia page for $\sigma$-algebra says this set is called a "sigma-algebra" by some, and called a "sigma-field" by others. I'm writing a paper on measure theory, where the topic of sigma-algebra ...
0
votes
2answers
46 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
160 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: ...