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
30 views

Finding normal basis for GF(q^m) over GF(q)

Could you kindly explain, how can one find a normal basis for GF$(3^6)$ over the GF$(3^2)$? As I understood, I should start with finding the polynomial in a form $$a(x^2) + (a^9)x + a^{81},$$ which ...
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0answers
11 views

Name of $\bar{F_E}$?

Let $E/F$ be a field extension. Define $\bar{F_E}:=\{\alpha\in E : \alpha \text{ is algebraic over } F\}$. What is the standard name of this? Fraleigh calls it 'the algebraic closure of $F$ in ...
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2answers
33 views

Factorization of polynomial in a complete field

Let $k$ be a finite extension of $\mathbb{Q}$ and $|\cdot|$ an absolute value on it (either Archimedean or not). Let $L$ be the completion of $k$ with respect to this value, and take any irreducible ...
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3answers
487 views

Why do they need commutativity in the proof?

In this proof they prove it for fields, but they say that the proof holds for a commutative ring with unity. But why do we need it to be commutative, and why do we need a unity? Why does it not hold ...
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1answer
35 views

Question about $[K:F]=[K:E][E:F]$ for fields $E,F,K$

This is probably a dumb question. We have a theorem that states If $E$ is a finite extension field of a field $F$, and $K$ is a finite extension field of $E$, then $K$ is a finite extension of ...
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0answers
24 views

Let F be a field and let a,b ∈ F with a not equal to b. Show that the polynomials f ( x ) = x + a and g ( x ) = x + b are relatively prime

Let F be a field and let $a,b ∈ F$ with $a$ is not equal to $b$ . Show that the polynomials $f ( x ) = x + a $ and $g ( x ) = x + b $ are relatively prime. Only things I know is : A commutative ring ...
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1answer
72 views

Complex number field: ''essentially'' unique?

I solved the following exercise but have trouble making sense of the result: If $\widetilde{\mathbb C}$ is another field of complex numbers and $\varphi : \mathbb C \to \widetilde{\mathbb C}$ is a ...
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2answers
25 views

If $x^3 + x + 1$ is reducible over $F$, then why does it follow that $E\subseteq F$?

I'm trying to understand a solution to the following problem: Show that $K = \mathbb Z_2[y,z]/\langle y^4 + y^3 + 1,z^3 + z + 1\rangle$ is a field. A solution is as follows (the part I don't ...
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0answers
50 views

A way to sum supernatural numbers involving Zeta function's analytic continuation

I have this idea on how to sum supernatural numbers assigning them a finite value in a way similar to how we assume that the sum of every natural numbers from 1 to infinity equals $-\frac 1 {12}$. ...
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2answers
804 views

Do the Liouville Numbers form a field?

The Liouville numbers are those which are better-than-polynomially approximated by rationals. More precisely, we say $x\in\mathbb{R}$ is Liouville when for all $n\in\mathbb{N}$ there is a $\tfrac ...
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1answer
39 views

A Field has characteristic 0 if and only if it contains Q

I would like some help in proving the following statement: A field $K$ has characteristic 0 if and only if $\mathbb{Q}$ is a subfield of $K$. So, the ...
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1answer
34 views

Inverting the elements of a basis of a finite field extension

Let $K/k$ be a finite field extension of degree $d$. Suppose that $\{a_1, \dots, a_d\}$ is a basis of $K$ as a $k$-vector space. Is it true that $\{a_1^{-1}, \dots, a_d^{-1}\}$ is a basis of $K$ as a ...
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1answer
45 views

Roots of a polynomial and finite additive subgroup

Suppose $F$ is a field with characteristic $p$ and $f(x)\in F[x]$ Then$f(x)=x^{p^m}+a_1x^{p^{m-1}}+\cdots +a_mx \iff$ its roots form a finite subgroup of the additive group of the splitting field. ...
2
votes
1answer
62 views

Transcendence Degree of a field extension over $\mathbb C$

Consider the $2 \times n$ matrix $\begin{bmatrix} x_{11} & x_{12} & x_{13} & \dots & x_{1n} \\ x_{21} & x_{22} & x_{23} & \dots & x_{2n} ...
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1answer
23 views

How is the degree of the minimal polynomial related to the degree of a field extension?

I was reading through some field theory, and was wondering whether the minimal polynomial of a general element in a field extension L/K has degree less than or equal to the degree of the field ...
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3answers
44 views

Find the values of $a,b,c$ such that $(1+\sqrt[3]{4})/(2-\sqrt[3]{2})=a+b\sqrt[3]{2}+c\sqrt[3]{4}$

Let $a,b,c\in\mathbb{Q}$. Find the values of $a,b,c$ such that $$(1+\sqrt[3]{4})/(2-\sqrt[3]{2})=a+b\sqrt[3]{2}+c\sqrt[3]{4}$$ What I tried: I multiplied both sides by $(2-\sqrt[3]{2})$. I still ...
0
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1answer
13 views

The field of rational functions of $k$ contains an algebraic closure of $k$?

If $k$ is a field and $k(\mathbf{x}) = k(x_1,\dots,x_m)$ is the field of rational functions in the indeterminates $x_1,\dots,x_m$, then if $f(y) \in k(\mathbf{x})[y]$ is irreducible, take the ...
3
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1answer
27 views

How many elements does $\mathbb{Z}_2(a)$ have, where $a$ is a zero of $f(x)$ in some extension field of $\mathbb{Z}_2$

Let $\mathbb{Z}_2=F$. Let $f(x)=x^3+x+1\in\mathbb{Z}_2[x]$. Suppose $a$ is a zero of $f(x)$ in some extension field of $\mathbb{Z}_2$. How many elements does $F(a)$ have and express each member of ...
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0answers
25 views

Does there exists an automorphism of $\Bbb{C}$ that's also an exponential hom?

Is there an automorphism of the field $\Bbb{C}$ of complex numbers, $\phi$, such that for all $z, w \in \Bbb{C}$ we have in addition to being a ring hom, $\phi(z^w) = \phi(z)^{\phi(w)}$?
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3answers
69 views

Showing that $\mathbb{Q}(\sqrt{3})$ isomorphic to $\mathbb{Q(\sqrt{-3})}$. (or possibly disprove it)

How can I show that $\mathbb{Q}(\sqrt{3})$ is isomorphic to $\mathbb{Q(\sqrt{-3})}$. (or possibly disprove it)? What I know: I don't know how to begin if it is the case that they are not isomorphic. ...
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2answers
71 views

Find the splitting field of $x^3-1$ over $\mathbb{Q}$.

Find the splitting field of $x^3-1$ over $\mathbb{Q}$. My try: Factoring this to the most I can (in $\mathbb{Q}$), we get that $(x-1)(x^2+x+1)$ So $x=1$ is a root of $f(x)$. $x^2+x+1$ has no ...
1
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1answer
48 views

If $\beta$ is a zero of $f(x)=x^2+x+2$ over $\mathbb{Z}_3$, find the other zero

If $\beta$ is a zero of $f(x)=x^2+x+2$ over $\mathbb{Z}_3$, find the other zero. What I tried: Suppose $B$ is a zero of $f(x)$, so $f(\beta)=\beta^2+\beta+2=0$. I know that $f(x)$ is irreducible in ...
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3answers
69 views

Why is an automorphism of $\mathbb R$ continuous

I was trying to understand this answer here but got stuck. It's clear to me that $\varphi: \mathbb R \to \mathbb R$ should map positive numbers to positive numbers and that it follows from that that ...
1
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2answers
50 views

Are the following real numbers constructible?

1) $\sqrt[4]{5+\sqrt2}$ 2)$\sqrt[6]{2}$ 3) $3/(4+\sqrt13)$ 4) $3+\sqrt[5]{8}$ From what I know, a number is constructible if it can be converted in a finite number of steps using only the ...
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2answers
32 views

Why does the automorphism mapping $\omega$ to 1, not an element of Galois group

Let $L/\mathbb{Q}$ be a field extension, where $L=\mathbb{Q}(\omega)$ and $\omega=e^{\frac{2\pi i}{7}}$. In my textbook it states that $Aut(L/\mathbb{Q})=\{\sigma_i|\sigma_i(\omega)=\omega^i, 1\leq ...
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1answer
20 views

Totally ordered non archimedian fields.

While thinking about properties of real closed fields, I came across the following contradiction: Let $(k,+,.,0,1,\leq)$ be a non archimedian totally ordered field. Let us assume that $(k,\leq)$ is ...
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1answer
49 views

About the construction of resolvents in Galois theory (over $\mathbb{Q}$ in $\mathbb{C}$)

I have to say that my question is quite long and I apologize for this. The main idea is that I would like to show how to construct resolvents for any transitive subgroup of the permutation group to ...
6
votes
1answer
31 views

Why are separable and normal field extensions so called?

To my understanding: A separable extension $K/F$ is one in which the minimal polynomial of every $\alpha\in K$ has no multiple roots. A normal extension $K/F$ is one in which some polynomial $f\in ...
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2answers
43 views

A universal construction of the field of fractions of an integral domain?

Let $R$ be an integral domain and For a field $\hat R$ consider the following : There is an injective ring homomorphism $i:R \to \hat R$ such that for any field $F$ and any injective ring homomorphism ...
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1answer
38 views

Prime ideal in the ring of integers of the number field $\mathbb{Q}(x)$ with $x^{3}=2$

In an exercise of the book Algebraic Theory of Numbers by Samuel, one must show that--in the integer ring $\mathcal{O_k}$ of the number field extension $\mathbb{Q}(x)$ where $x^{3}=2$--the ideal ...
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1answer
20 views

Simplifying the Splitting field of $x^n-a$

Let $L/K$ be a field extension where $L$ is the splitting field of the polynomial $x^n-a\in K[x]$. Clearly $L=K(t,\zeta t,\ldots,\zeta^{n-1}t)$, where $t=\sqrt[n]{a}$ and $\zeta$ is the primitive ...
2
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1answer
44 views

(non Galois) correspondence

If $L/K$ is a field extension we note $\textrm{Aut}_K (L)$ the group of field automorphisms of $L$ that are fixing each element of $K$. Fix an extension $L/K$ and note $G:=\textrm{Aut}_K (L)$. If $H$ ...
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1answer
30 views

Is it true that the splittings fields cannot be isomorphic?

Let $F$ and $K$ be any two different fields and $p$ be a prime number. Let $X^q-X\in F[X]$ and $X^q-X\in K[X]$ where $q=p^k$ with integer $k>1$. Are the splitting fields of these polynomials ...
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0answers
22 views

How to formally show a field extension is not radical

I'm wondering if there is a general procedure for showing that a field extension is not radical. As an example, let $L=\mathbb{Q}(\sqrt[3]{1+\sqrt{2}})$. Then I can see that $L/\mathbb{Q}$ isn't a ...
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2answers
116 views

Around a little mistake in Eisenbud's Commutative Algebra: What does $k(x)\otimes k(x)$ look like?

In Prof. Eisenbud's Commutative Algebra with a view ... book, in Appendix A1 p. 564 in his proof of a result of Maclane he said that if $L$ is an extension of a field $k$, we have $L\otimes_k ...
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1answer
26 views

Norm of an ideal is finite

I want to show that the norm $N_{K/\mathbb Q}(\mathfrak{a})$ of $\mathfrak{a}$ a nonzero integral ideal of a number field $K$ is finite, and so $N_{K/\mathbb Q}(\mathfrak{ab})=N_{K/\mathbb ...
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2answers
42 views

Elementary questions about polynomials and field extensions

Let $$f(x)=x^2+x+1.$$ This is irreducible in $\mathbb{Z_2}[x]$, and thus $\mathbb{Z_2}[x]/(f(x))$ is a field $K$ where $(f(x))$ is a principle ideal. I don't quite understand how I find that ...
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0answers
24 views

In a Euclidean domain, show that for nonzero $a,b \in D$, $v(a) < v(ab)$ iff $b$ is not a unit of D.

The function $v(x)$ is a Euclidean function on an integral domain, D. Proof : Suppose that $v(a) < v(ab)$. If $b$ were a unit, then $a$ and $ab$ would be associates. We have $a = abu$ and $ab = ...
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1answer
213 views

A shortcut in Galois theory

How could we prove Galois correspondence without using Dedekind’s Lemma on group characters, Artin’s lemma and the primitive element theorem ? I just came across Meinolf Geck's article On the ...
5
votes
3answers
95 views

A finite field extension of $\mathbb R$ is either $\mathbb R$ or isomorphic to $\mathbb C$

Let $F$ be a field containing $\mathbb R$ with the property that $\dim_{\mathbb R}F < \infty.$ Then either $F \cong \mathbb R$ or $F \cong \mathbb C.$ I am trying to prove the above statement. ...
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1answer
50 views

Show for an irreducible polynomial $f(x) \in F[x]$ of degree $n$, $n$ divides $[E:F]$ where $E/F$ is the splitting field of $f(x)$

I want to show that for an irreducible polynomial $f(x)$ in $F[x]$ of degree $n$, and for a splitting field $E/F$ of $f(x)$, $n$ $|$ $[E:F]$. Can anyone provide any hints for me to think about this? ...
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votes
4answers
57 views

For $F[\alpha]$ as a finite field extension of $F$, is $1/\alpha \in F[\alpha]$?

Is the multiplicative inverse of $\alpha$, that is, (1/$\alpha$) contained within $F[\alpha]$? Thanks in advance.
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4answers
49 views

$\mathbb {Q}(\sqrt{2},\sqrt{6}) = \mathbb {Q}(\sqrt{2},\sqrt{3})$

$$\mathbb {Q}(\sqrt{2},\sqrt{6}) = \mathbb {Q}(\sqrt{2},\sqrt{3})$$ I know this may seem trivial but could someone please explain the logic behind this statement to me? Thanks in advance.
0
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1answer
27 views

Algebraically Independence of polynomials.

Are the polynomials $(xy+xt+zt)t$, $(x+z)t^2$, $(x+z)(y+t)t$, $(y+t)(xy+xt+zt)$ algebraically independent ? If not what are all the relations between them. I tried to compute the determinant of the ...
4
votes
1answer
61 views

Difference between F[x] and F(x)

Notation wise, what is the difference between F[x] and F(x)? Is F[x] the ring of polynomials with coefficients in F, and F(x) the field of rational functions with coefficients in F? I am asking ...
5
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1answer
82 views

Is the sum of an algebraic and transcendental complex number transcendental?

I was wondering if the sum of an algebraic and transcendental complex number is transcendental. I was thinking if a is algebraic, and b is transcendental, then if a+b is algebraic, then a+b-a is also ...
0
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1answer
11 views

How does cardinality affect the dimension of an extension field?

As an example: Let p be an odd prime. When α, β ∈ $\mathbb{Z}$p are not squares, then α / β is a square. Let L, K both be fields of size p2. The rest of the exercise is irrelevant to what I ...
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0answers
38 views

why is $ F [x]/(x^2 -1)$ isomorphic to $ F \times F $ [duplicate]

why is $ F [x]/(x^2 -1)$ isomorphic to $ F\times F $ I know F is a field where 1+1 can't equal 0 I've calculated the idempotents as $1/2 x+1/2$ and $-1/2 x+1/2$ not sure what to do next
0
votes
1answer
40 views

Is this mapping an isomorphism?

Define $\phi:\mathbb Q[\sqrt3]\rightarrow \mathbb Q[\sqrt7]$ by $\phi(a+b\sqrt3)=(a+b\sqrt7)$. Is $\phi$ isomorphic? Is there any isomorphism at all? So I started this by writing for the given a,b ...
1
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0answers
16 views

Finding ring isomorphism [duplicate]

Find isomorphism from $F_5[x]/(x^2+x+2) \rightarrow F_5[x]/(x^2+4x+2) $ I realise both polynomials are irreducible therefore form fields, not sure how to form a isomorphism from one to the other, help ...