0
votes
1answer
26 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 ...
1
vote
0answers
83 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 $f(x) = x^n-x$ and $f'(x) = nx^{n-1}-1$ If $f(x)$ has a multiple zero, then, $f(x)$ and $f'(x)$ have a common factor. An irreducible ...
0
votes
2answers
48 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
38 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
34 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
92 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
92 views
+100

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, ..., ...
3
votes
0answers
67 views

What is the Difference Between an Algebra and a Field?

The Wikipedia page for $\sigma$-algebra, as well as some other resources I'm studying, say this set is called a "sigma-algebra" by some, and called a "sigma-field" by others. I'm writing a paper on ...
0
votes
2answers
42 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
141 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
35 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
22 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, $p$. ...
1
vote
2answers
48 views

What are subfields of $\mathbb{C}$?

I only took the first undergraduate abstract algebra course, so i don't know (at all) what Galois theory is about. I'm asking this question since i'm not sure of the definition of inner product space ...
2
votes
1answer
30 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 $F(a)$ ...
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 ...
3
votes
3answers
92 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 ...
1
vote
2answers
50 views

There always exists a subfield of $\mathbb C$ which is a splitting field for $f(x)$ $\in$ $Q[X]$?

So I've been studying field theory on my own, and I just started learning about splitting fields. Based on my understandings if a polynomial, $f(x)$ $\in$ $Q[X]$, then there should be always a ...
2
votes
2answers
29 views

Transitivity Property of Separable Extensions

I was looking for some proof for the transitivity property of separable field extensions. Although this might sound like a very well-known fact and is referred to frequently, I do not seem to find a ...
1
vote
3answers
27 views

Quadratic number fields containing primitive roots of unity

A problem from Michael Artin's Algebra (Second Edition) from Fields: Determine the quadratic number fields $\mathbb{Q}[\sqrt{d}]$ that contain a primitive $n$th root of unity, for some integer $n$. ...
2
votes
2answers
74 views

How do elements of $\mathbb{R}(xy,x+y)$ look like?

I have problems with determining what are typical elements of such field $\mathbb{R}(xy,x+y)$ In one indeterminate it is easier as $\mathbb{R}(x)=\Bigl\{\frac{f(x)}{g(x)}, g(x)\neq 0, ...
1
vote
1answer
47 views

Describe the elements in $Q(\pi)$

Describe the elements in $Q(\pi)$ Attempt: $Q(\pi)$ is the smallest field which contains $Q$ and $\pi$ We know that $\nexists~ f(x) \in Q[x]$ such that $f(\pi)=0$ Hence, $Q[x]/\langle p(x) ...
0
votes
2answers
30 views

Zeroes of f(x) in a splitting field $E $ have the same multiplicity

Let $f(x)$ be an irreducible polynomial over a field $F$ and let $E$ be a splitting field of $f(x)$ over $F$. Then all the zeroes of $f(x)$ in $E$ have the same multiplicity. The proof of this ...
0
votes
2answers
33 views

Discrete valuations of the rational numbers

I'm trying to find every discrete valuation on the field of rational numbers. If $a\in \mathbb Q$, we can write $a=p^j\frac{x}{y}$, where $p$ is a prime number and $p\nmid x$ and $p\nmid y$. We can ...
2
votes
1answer
36 views

Let $F$ be a field and let $f(x)$ be a non constant element of $F[x]$. Then, there exists a splitting field $E$ for $f(x)$ over $F$.

Let $F$ be a field and let $f(x)$ be a non constant element of $F[x]$. Then, there exists a splitting field $E$ for $f(x)$ over $F$. I have some queries regarding this theorem of existence of ...
2
votes
3answers
134 views

Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$

Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$ I have asked a similar problem Minimal Polynomial of $\zeta+\zeta^{-1}$ and i tried to repeat similar idea ...
2
votes
1answer
41 views

Places of this extension

I'm reading this book. I'm trying to find the degree of the places of the extension $\mathbb C(X)\mid\mathbb R$. I know the places of the extension $\mathbb R(X)\mid\mathbb R$ and I've already ...
2
votes
4answers
140 views

Is there a ring $K$ such that $\mathbb R\subsetneqq K\subsetneqq \mathbb C$?

Is there a ring $K$ such that $\mathbb R\subsetneqq K\subsetneqq \mathbb C$? and if $K$ is a local ring, is there such a ring? I need this result be negative to prove a question I'm working. I need ...
2
votes
1answer
33 views

Isomorphic algebraic closures.

I've just stared learning the Galois Theory so my question might be trivial, but could someone give me an example of two different algebraic closures of the same field? Cause I don't get how they can ...
1
vote
1answer
47 views

Confusion regarding proof of a proposition in Field Theory (Dumb Question)

Question is to prove that : For finite extensions $E/k$ and $F/k$ Prove that $[EF:k]\leq [E:k][F:k]$ where $EF$ is the smallest field extension which contains both $E$ and $F$ Supposing ...
2
votes
1answer
24 views

Sources about transcendence degree

I asked this question: Characterization of the transcendentals over a field I realized I need some knowledge about transcendence degree to prove some facts in the book I'm reading. I would like to ...
0
votes
0answers
77 views

Query on an example of Morandi's Field and Galois Theory, regarding the degree of a field extension.

I am going through Morandi's Field and Galois Theory, and I am looking at Example $1.5$, Chapter I. It says, (more or less) If $k$ is a field, let $K=k(t)$ be the field of rational functions in ...
1
vote
3answers
44 views

Extensions of degree $1$.

My doubt is very simple: Let $F|K$ be a field extension, if $[F:K]=1$, what can we say about $F$ and $K$? can I say $F=K$? I'm trying to prove the equality without success. Thanks in advance
4
votes
1answer
65 views

Milne's Galois Theory Example

The following example is drawn from Milne's Galois Theory notes, p.42 (http://www.jmilne.org/math/CourseNotes/FT.pdf) We study the extension $\mathbb{Q}[\zeta]/\mathbb{Q}$ where $\zeta=e^{2\pi i/7}.$ ...
3
votes
1answer
59 views

real closure of an archimedean field

my question is: Is an archimedean field dense in its real closure? I know that in the non-archimedean case, this does not have to be true (e.g., rational fucntions). Thanks!
2
votes
1answer
49 views

Proving that a field $K$ can be generated by algebraically independent elements and an separable element

Let $k$ be a perfect field (either $k$ has characteristic $0$, or characteristic $p > 0$ and every element has a $p$th root), and let $K$ be a finitely generated extension field. I have a question ...
1
vote
1answer
25 views

Irreducibility in Galois/non Galois Extensions

Let $k$ be a field and $\alpha$ algebraic over $k$. Let $K$ be the Galois closure of $k(\alpha)$ (obtained by adding all conjugates of $\alpha$). If $f(x) \in k[x]$ is irreducible over $k[\alpha]$ is ...
2
votes
1answer
84 views

Projective special linear groups are co-hopfian?

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
0
votes
1answer
48 views

Additive inverse in a field is unique

there will be an $a$ let's prove $-a$ is unique. let's assume there are 2 additive inverses $b$ and $c$ therefore $a+b=a+c$ let's multiply them by the multiplicative inverse of $a$ I try to use the ...
2
votes
1answer
47 views

A Fixed Field of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$

I was working on a review problem from Dummit and Foote and came across the following issue. It is clear that the Galois group of the splitting field for the polynomial $(x^2-2)(x^2-3)(x^2-5)$ has ...
6
votes
5answers
303 views

A finite commutative ring with 1 whose elements satisfy a particular equation

I would be very grateful if you give me a hint on it: Suppose $R$ is a finite commutative ring with identity such that $ x^3 = x $ for all elements $x$ of $R$. Then $R$ is a finite direct product ...
1
vote
2answers
36 views

$K \le B\le F$, If $F$ Galois over $K$ $\Rightarrow$ $F$ a Galois over $B$

Let $F$ a Galois over $K$, and let $B$ be a subfield of $F$ such that : $K \le B\le F$ $\Rightarrow$ $F$ a Galois over $B$ PROOF: $F$ is a splitting field of $f \in K[x]$ separable over $K$. ...
5
votes
1answer
54 views

$F/K$ algebraic and every nonconstant polynomial in $K[X]$ has a root in $F$ implies $F$ is algebraically closed.

Let $F/K$ be an algebraic extension of fields in characteristic zero. If $F/K$ is normal, and every nonconstant polynomial $f \in K[X]$ has a root in $F$, then $F$ is algebraically closed. This is ...
5
votes
3answers
186 views

A question about algebraically closed fields

A field $\mathbb{K}$ is said to be algebraically closed in practice if every polynomial over $\mathbb{K}$ of positive degree less than or equal to $10^{10}$ has zero belonging $\mathbb{K}$. The ...
1
vote
0answers
44 views

Find a multiplicative inverse of an element in a field

Suppose we have an element $\sigma=p+qa\rho+rd\rho^{-1}\in K$ where $K=\mathbb{Q}(\rho)$ where $[K:\mathbb{Q}]=3$ I want to find a multiplicative inverse of $\sigma$ i .e ...
0
votes
3answers
58 views

For extension fields, does $[F(a,b):F(a)]=[F(b):F]$?

sorry if this question seems obvious, For a field F and $a,b\notin F$, does $[F(a,b):F(a)]=[F(b):F]$? If so, how do you prove it or is there a counter example?
1
vote
2answers
97 views

Existence of Irreducible polynomials over Z of any given degree which do not satisfy the Eisenstein's Criterion

I just came across the following interesting question which has been once discussed: Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree I was wondering if we could find such ...
3
votes
3answers
99 views

Is it possible to describe $Q(x)$ as the extension field of $R$ freely generated by $\{x\}$?

Given an integral domain $R$, the polynomial ring $R[x]$ can be defined as the commutative $R$-algebra freely generated by $\{x\}$. Also, let $Q$ denote the field of fractions associated to $R$. Then ...
3
votes
1answer
71 views

Galois closure of $\mathbb{C}(T,\sqrt{T^2+T+1})$ over $\mathbb{C}(T^3)$

I'm trying to solve the following problem. But it's too difficult for me. Let $\mathbb{C}(T)$ be a function field, and put $L:=\mathbb{C}(T,\sqrt{T^2 + T +1})$, $K:=\mathbb{C}(T^3)$. Let $M$ be a ...
1
vote
1answer
69 views

Solvability of polynomials over fields of characteristic zero

1) Let $K$ be a field, $\operatorname{char}(K)= 0$, and $f ∈ K [x]$ with $\deg(f)\le4$. Then $f$ is solvable by radicals. Proof: $\operatorname{Gal} (F/K) \cong S_4$ then $\operatorname{Gal}(F/K)$ ...