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
97 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
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2answers
88 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 ...
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2answers
153 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
37 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
62 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 ...
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1answer
25 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))+\\ ...
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3answers
146 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.
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1answer
118 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
45 views

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

How can I find $K$-automorphism $\sigma \in Aut(K(x))$ different from identity such that $\sigma (x(x+1))=x(x+1)$?
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1answer
89 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
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0answers
76 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 ...
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1answer
32 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
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2answers
543 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( ...
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1answer
146 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 $F$?
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1answer
196 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 ...
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0answers
47 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 ...
5
votes
1answer
115 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
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1answer
49 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
91 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 ...
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3answers
103 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 $F(x) \subset E(x)$ ...
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1answer
73 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 ...
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0answers
14 views

Closure of multiplication of Cong. classes

In mod p, let k be the smallest positive integer such that [a]^k = [1]. Show that {[a],[a]²,.....,[a]^k} is closed under multiplication. This is part of a larger problem overall and I have no idea ...
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2answers
86 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
48 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
150 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
33 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 ...
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2answers
253 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 ...
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votes
2answers
85 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 ...
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1answer
34 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
54 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 ...
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0answers
44 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 ...
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2answers
248 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, ...
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0answers
102 views

Is there a typo in this proof about Galois theory in Artin's Algebra?

The following is a statement in Artin's Algebra (2nd edition p. 489): Corollary 16.6.5 (a) Every finite extension $K/F$ is contained in a Galois extension. (b) If $K/F$ is a Galois extension, ...
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1answer
54 views

How do I find a basis for $\mathbb{Z}_5[x]/\langle x^3-x^2-1 \rangle$?

I wish to find a basis for the field $\mathbb{Z}_5[x]/\langle x^3-x^2-1 \rangle$. Treating the polynomial as the additive identity, my intuition tells me that it should be $\{1,x,x^2\}$. Thus, finding ...
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0answers
192 views

A question about the isomorphism extension theorem.

My book (Fraleigh) states the following: Assumption: All algebraic extensions of $F$ are assumed to be contained in a fixed algebraic closure $\overline{F}$ of $F$. Isomorphism Extension ...
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0answers
34 views

Question on finite fields

I was curious about this question: Let $p$ be a prime, and $d \geq 1$ and $K$ is a field of $p^d$. How many proper subfields does $K$ have? All I know if that a finite field has order $p^n$, where ...
2
votes
1answer
59 views

How can I prove that $[\mathbb{Q}(\omega + \omega^{-1}) :\mathbb{Q}]=\frac{\varphi (n)}{2}$? [closed]

If $n>2$, $\omega \in \mathbb{C}$ an $n$-th primitive root of unity then $[\mathbb{Q}(\omega + \omega^{-1}) :\mathbb{Q}]=\frac{\varphi (n)}{2}$. ($\varphi (n)$ is the Euler totient function.) ...
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2answers
738 views

Proofs field axioms

I'm terrible with these kinds of proofs and this might be a duplicate but I didn't find anything. I need to prove that $\forall x,y \in \mathbb{R}: -(-x)=x$ and $-(x+y) = -x - y$ I know that this is ...
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1answer
97 views

Irreducibility over $ \mathbb{Q} ( \sqrt{2} , \sqrt{3})$ [closed]

Show that $x^5-9 x^3 +15x +6$ is irreducible over $ \mathbb{Q} ( \sqrt{2}, \sqrt{3})$
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1answer
38 views

A question about finite fields with order $p^n$.

If we have a finite field $E$ with $p^n$ elements, where $p$ is a prime number. Is it necessary that it is a field extension of a subfield isomorphic to $\Bbb{Z_p}$, and consequently is of ...
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2answers
47 views

Degree of an element in a Field Extension

If $\alpha \in \overline{F}$ s.t. $\alpha^2 = s$ for some $s \in F$ can we say that degree of $\alpha$ over $F$ equals $2$?
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2answers
163 views

Continuity and Inverse Preservation of Boundedness Implies Preservation of Closed Sets

Sorry about the rather long title - wasn't sure what else to call it! Here is my question: Let $f : \Bbb R^n \rightarrow \Bbb R^m$ be continuous and such that $f^{-1}(F)$ is bounded whenever $F$ ...
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Does “K” a Field?

Does the group $$ K =\begin{matrix}\pmatrix{ a & 0 \\ 0 & 0\\ } \end{matrix} $$ is a Field relatively to additive and multiply methods about Matrices? I tried to show that it's NOT by ...
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0answers
56 views

existence of purely inseparable extension

Let $F<E$ be a finite extension that is not separable. Show that for each $n\geq 1$, there exists a subfield $E_n$ of $E$ for which $E_n<E$ is purely inseparable and $[E:E_n]_i=p^n$ ($[...]_i$ ...
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1answer
53 views

Field of Quotients Explanation

I'm having a hard time grasping the concept of a field of quotients. The book I'm currently reading gives the following definition: Any integral domain D can be enlarged to a field F such that every ...
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0answers
48 views

separable degree and the radical exponent

Let $\alpha$ be algebraic over $F$, where $charF=p\neq0$ and let $d$ be the radical exponent of $\alpha$. (which means $\alpha$ has multiplicity $p^d$) I am trying to show the following expression; ...
3
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2answers
420 views

Galois Group for $x^5-1$

This question is an extension to the question in math.stackexchange.com/questions/759230/subfield-of-the-galois-group-of-x5-1 It seems the discussion in that topic is dead and I still have a major ...
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1answer
67 views

A question of algebraic geometry applied to field theory

I’ve come across this question in a coding theory course, and it has stumped me. Any hints and/or suggestions would be appreciated. Let $F$ be a field (for our purposes, assumed to be finite of ...
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3answers
47 views

Find values of $p$ for which in a field $\mathbb{Z}_p$, two equations have 1 solution.

Find values of $p$ for which in a field $\mathbb{Z}_p$, two equations, say $7x-y=1$ and $11x+7y=3$ have 1 solution. I can give some values of $p$ like the obvious $p = 7, 11$. But how do I ...
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2answers
101 views

Galois Group Calculation

Calculate the Galois Group $G$ of $K$ over $F$ when $F=\mathbb{Q}$ and $K=\mathbb{Q}\big(i,\sqrt2,\sqrt3 \big)$. My thoughts are as follows: By the Tower Lemma, we can see that ...