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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1answer
27 views

Automorphism group of a non-normal field extension

Consider finite field extensions $L>K>F$ such that $L/F$ is Galois, and $K/F$ is separable. I am particularly interested in the case $F=\mathbb{Q}$. By Galois theory, $K/F$ is normal iff ...
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3answers
67 views

Prove that every extension of a finite field is normal

In book by Roman 'Field Theory' it is written that it is straightforward that every extension of a finite field is normal. However I just cannot see it. Can you help me with this problem? Thank
8
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2answers
101 views

Order of invertible matrices

I recently came across an interesting problem in Artin which says: If $A \in GL_2(\mathbb{Z})$ is of finite order then it has order $1,2,3,4,6$. I was looking for a generalization of this problem. ...
5
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1answer
68 views

$Gal(\mathbb{Q}(\sqrt 2 + \sqrt 3)/\mathbb{Q})$

A basis for $\mathbb{Q}(\sqrt 2 + \sqrt 3)$ over $\mathbb{Q}$ is $\{1,\sqrt 2 , \sqrt 3 , \sqrt 6 \}$ The roots of $x^2 -2$ are $\pm \sqrt 2$ and the roots of $x^2 -3$ are $\pm \sqrt 3$ so to find ...
0
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1answer
52 views

finite field extension and normality

If $K$ is an extension field of $F$ such that $[K:F]=2$. Then $K$ is normal? I know that if $[K:F]=2$ then $K=F(u)$ where $u$ is the root of $f \in F[x]$. But how do you prove that dimension $2$ ...
4
votes
1answer
42 views

Why is the order of a prime element well-defined?

This is in relation to the $p$-adic valuation on the field of fractions $F$ of an integral domain $D$. The idea is that for each $x \in F$ there is a unique maximal $k$ such that $x = p^k u v^{-1}$ ...
1
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1answer
34 views

Rupture field of $X^p+T$ equals its splitting field [closed]

Let $K$ be a field of prime characteristic $p$. Let $P(X)=X^p+T$ be a polynomial from $K(T)[X]$. $P$ is irreducible over $K(T)$ by Eisenstein criterion. Show that a rupture field of $P$ is also a ...
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2answers
73 views

Field Extension Question: $K(a,b) \ne K(a+b)$

I am trying to come up with an example of when an extension $K(a,b)$ with $a,b$ in $E$ is not equal to $K(a+b)$. In short, $$K(a,b) \ne K(a+b).$$ I am learning abstract algebra for the first time, ...
1
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0answers
28 views

Reference for Galois Theory of infinite field extensions.

I would like to ask what is in your opinion the best place to learn about Infinite Galois Theory that requires not much knowledge of topology. I am searching for a text that explains the notions ...
4
votes
0answers
67 views

Isomorphism between finite fields adjoining a root

Let $p(x)=x^3+x^2+1$ and $q(x)=x^3+x+1$ be polynomials over the field $\mathbb{Z}_2$. Let $\alpha$ be a root of $p(x)$ and $\beta$ be a root of $q(x)$. Now let $K=\mathbb{Z}_2(\alpha)$ and ...
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0answers
15 views

Consider $n$ a product of distinct primes $p_j$, with each $p_j-1$ a power of 2, then the regular $n$-gon is constructible.

I am having trouble showing that if $n$ is a product of distinct primes $p_j$, with each $p_j-1$ a power of $2$, then the regular $n$-gon is constructible. Apparently it could be useful to use that ...
2
votes
1answer
42 views

Embedding a finite extension of $F(X)$ into a pure transcendental extension

If $F$ is any algebraically closed field, and $L \supset F(X)$ is a finite extension of the purely transcendental extension of $F$ of transcendence degree $1$, then can $L$ necessarily be embedded ...
2
votes
1answer
62 views

Structure of a module and finite fields

I am trying the following problem which appeared on an exam for a grad course: Determine the structure of finite field $F$ as a module over $F_p[x]$ when $xv=Lv$. Here $L(z)=z^p$ and ...
0
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1answer
16 views

a question about normal extensions

Let $E$ be a finite normal extension of the field $F$, for which the Galois group $G(E/F)$ is Abelian, and let $K$ be a field intermediate to $F$ and $E$. Let $a\in K$ and $f(x)\in F[x]$ be the ...
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1answer
79 views

What is zero times zero

What is zero zeros? What are no nothings? From a mathematical point of view it would be my thing, but none of us are educated that much in math, so I am curious to hear an expert opinion.
1
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1answer
34 views

Prove that $B$ is a subfield of $F$

If a subring $B$ of a field $F$ is closed with respect to multiplicative inverses, then $B$ is a field. Fields are commutative rings with unity, and every nonzero element has an inverse. A subring ...
2
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0answers
31 views

Number of elements and number of different basis of $\mathbb F_5^3$

Let $\mathbb F:=\mathbb F_5$ the field with five elements. (i) How many elements has $\mathbb F^3$? (ii) How many different basis has $\mathbb F^3$? My idea: (i) $\mathbb F^3$ has $5^3$ elements. ...
0
votes
1answer
21 views

Basis for field extension by an algebraic element

Is was wondering if, given a field F with a known basis and an element b which is algebraic over that field, it is possible to construct explicitly a basis for F[b], the extension of F by b. Suppose, ...
2
votes
1answer
73 views

Why isn't $e^{\frac{2\pi i}{9}}$ an element of $\mathbb{Q}(\sqrt[9]{2}, e^{\frac{2\pi i}{3}})$?

The question is in the headline above. I need to know this, because I want to show that $\mathbb{Q}(\sqrt[9]{2}, e^{\frac{2\pi i}{3}})/\mathbb{Q}$ is not a normal extension and it's quite obvious I ...
1
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1answer
53 views

Choosing a polynomial for CRC

CRC checksum is a homomorphism from polynomials over $\mathbb F_2$ to itself. As I understand, the map $f\mapsto g$ it is simply remainder from division $f$ by $p$, where $p$ is a fixed polynomial for ...
2
votes
2answers
39 views

Polynomial ring and extension field

Let $K$ be a field and $p(x) \in K[x]$ a monic irreducible polynomial of degree $n$, suppose $F/K$ is a field extension, and there exist $u \in K[x]$ which is a root of $p(x)$. 1) Let $K(u)$ be the ...
0
votes
2answers
70 views

Calculate the degree of a field extension

Find degree $[\mathbb{Q}(i,\sqrt{2}) : \mathbb{Q}]$ let $a = \sqrt{2}$ $a^2 = 2$ $\therefore a$ is a root of $q(x) = x^2 -2$, where $q(x)\in\mathbb{Q}(i\sqrt{2})[x]$ means degree of $a$ over ...
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1answer
32 views

Proving that operations give equal results given equal inputs

I was reading about the 9 or 12 basic properties of 'fields' (if that's what they're called) in a book called Spivak's Calculus, 3rd Edition, and got quite befuddled by dealing with as basic stuff as ...
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1answer
37 views

Show that two field extensions are the same

Can you help me with showing that these two field extensions are the same: $\mathbb{Q}(\sqrt{3}, \sqrt[3]{5})$ $\mathbb{Q}(\sqrt{3} + b\sqrt[3]{5})$, where $b\neq0$ is any rational number. Thanks ...
2
votes
1answer
17 views

Computing determinants using derivatives in an arbitrary field

When computing determinants that depend on a parameter $t\in \Bbb R$, it is often useful to use the fact that \begin{align} \det(V_1(t),\dots,V_n(t))&=\det(V_1(a),\dots,V_n(a))+\\ ...
3
votes
3answers
127 views

Check that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational

How to prove that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational. I will appreciate any proof, but I had such exercise during lecture in field theory. Thanks.
1
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1answer
48 views

Find a subfield of $\mathbb{C}$ isomorphic to other field

Do you know, how I can find a subfield of $\mathbb{C}$ isomorphic to $F = \mathbb{Q}(\sqrt[3]{7})$ such that $F \nsubseteq \mathbb{R}$? I don't even have clue, how I should start. Thanks
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1answer
38 views

nontrivial $K$-automorphism of $K(x)$

How can I find $K$-automorphism $\sigma \in Aut(K(x))$ different from identity such that $\sigma (x(x+1))=x(x+1)$?
3
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1answer
53 views

Is there a nice topology on Aut(C)?

Let $G=\mathrm{Aut}(\mathbf C)$, the group of field automorphisms of the complex numbers. It is a very large group (see this MSE question and the nice answer by Andres). For instance, there even ...
3
votes
0answers
20 views

A notion of transcendence degree for fields of formal power series?

There is an intuitive sense in which one would like to say that the field of formal Laurent series $k((z))$ over a field $k$ has "transcendence degree $1$". Of course, it doesn't have transcendence ...
1
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1answer
30 views

A doubt regarding splitting field.

$\Bbb Q(\omega)= \Bbb Q(\sqrt3,\iota)$ This is written in my text book that i am following. But I think this is a typo. Since $\Bbb Q(\sqrt3,\iota)$ is a larger field. in which $\Bbb Q(\omega)$ is ...
12
votes
2answers
517 views

Is number rational?

How can we check if number $a=\frac{ \sqrt[4]{2}+\sqrt[3]{3}}{\sqrt[4]{2}+\sqrt[3]{3} +1}$ is rational? Is there any smart solution? Another assignment is to find $\left( ...
0
votes
0answers
21 views

Karatsuba Method

For polynomials $f(x)$, $g(x)$ of degree $d = 2^{r-1}-1$, how do I check that multiplying $f(x)$ and $g(x)$ by the Karatsuba method requires $3^{r-1}$ multiplications in the field $F$?
0
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1answer
56 views

The Galois Group of $x^4 - 5x^2 + 6$

As the title suggests, I'm asked to describe the Galois group of the polynomial $x^4 - 5x^2 + 6 \in \mathbb{Q}[x]$ over $\mathbb{Q}$. I am pretty certain I have 95% of the problem completed. I'm just ...
1
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0answers
32 views

Function Field of Degree 3 Ramified at 1 and -1

This question is a homework problem, and I'm having a lot of trouble with it. (a) Determine the number of isomorphism classes of function fields K of degree 3 over $F = \mathbb{C}(t)$ that are ...
4
votes
1answer
31 views

Finding the conjugates in $\mathbb{C}$ of a given number over a given field…

I'm having somewhat of a difficult time understand what's being asked—and thus having a hard time answering the question: Find all conjugates in $\mathbb{C}$ of the given number over the given ...
0
votes
1answer
41 views

Does $E$ a finite field and $F\subset E$ imply that $E$ is Galois over $F$?

Is this the case? I don't know whether to go fishing for a counterexample or to try to prove it.
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0answers
48 views

How many different proper subfields does $K$ have, where $K$ is a field of order$p^n$?

If $p$ is a prime, and $d$ is an integer greater than or equal to 1. Let $n= p^d$, then there exists $K$ being a field of order $p^n$. But how many different proper subfields does $K$ have? The ...
1
vote
3answers
87 views

Field extensions - if $(E : F) = n$ then $(E(x) : F(x)) = n$

Well, pretty much everything is in the title - I'm looking for the proof of the following statement: if we have a field extension $F \subset E$ then the degree of the extension $F(x) \subset E(x)$ ...
0
votes
1answer
30 views

Convert from a field extension to an elementary field extension

I have a basic question about algebraic field extensions: How can I convert a multiple extension like $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to a single (elementary) field extension (like ...
1
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2answers
48 views

show the splitting field of polynomial

Show that $\mathbb{Q}(\sqrt[4]{2}, i)$ is also the splitting field of $x^4 + 2$ over $\mathbb{Q}$. I solve it as $x^4+2=0$ then $x^4=-2$ $\implies \mathbb Q(\sqrt[4]{-2}, \sqrt{-2})$
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1answer
23 views

can we have split field of the real number and how i get the extension dimension?

what the spilt field of x^3-8? I think I cannot split the real number,is that correct? what about x^3-2 ,is the extension dimension will be either 6 or 3? In the beginning I think it will be 3 ...
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0answers
34 views

Topology in Infinite Galois Theory.

I am a final year undergraduate student in Mathematics. I have a good background in algebra up to Galois theory of finite extensions of fields. I have started trying to understand the Galois theory of ...
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0answers
30 views

$G_f^\theta$ is $A_4$ or $S_4$?

Let $f(x)\in \mathbb{Q} [x]$ irreducible polynomial of degree 4, $u \in \mathbb{C}$ a root of $f(x)$. Prove that there are not subfields $K$ such that $\mathbb{Q} \subset K \subset \mathbb{Q} (u)$ if ...
1
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2answers
71 views

Irreducible Polynomials over a Finite Field

I am motivated by this question, and want to find a solution to the following problem. Question: How many monic, irreducible polynomials of degree $n$ are there over the finite field ...
0
votes
2answers
76 views

A solution to the equation $\frac{1}{x}=0$ [duplicate]

The number $i$ is defined as a solution to the equation $x^2+1=0$. How come no one has yet defined a number $j$ as a solution to the equation $\frac{1}{x}=0$? The purpose of course is to be able to ...
0
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1answer
28 views

Field Theory question

This question does not make sense to me. For one I believe it does not tell us if $\gamma$ is in our field $E$. For all we know $\gamma$ and for that matter $\xi$ can have a degree of $1$ or $2$. ...
3
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0answers
30 views

Minimal polynomial for sum of algebraic numbers. [duplicate]

If I have two algebraic numbers $\alpha,\beta$ such that $A(\alpha) = 0$ and $B(\beta)=0$ where $A,B \in \mathbb{Q}[x]$ are the minimal polynomials of $\alpha$ and $\beta$ respectively. Knowing only ...
0
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0answers
40 views

Check that this is a category

Assume we have a fixed field $F$. We define objects as homomorphisms $\phi:F\rightarrow G$. Then we define morphisms between $\phi:F\rightarrow G$ and $\psi:F\rightarrow L$ as ring homomorphism from ...
3
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2answers
95 views

Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?

As title, Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated? Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, ...