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 ...
1
vote
2answers
45 views
Show $\zeta_p \notin \mathbb{Q}(\zeta_p + \zeta_p^{-1})$
I asked a question here: [Writing a fixed field as a simple extension of $\mathbb{Q}$ ], but realised I couldn't justify why the given quadratic was irreducible.
Thus: Is there a way of showing ...
1
vote
1answer
38 views
characteristic vs minimal polynomial
Let $L$ be a finite field extension of $K$. For every element $\theta$ in $L$ define the characteristic polynomial of $\theta$ as follows
$$\operatorname{char}_{\theta}(X):=\det(X\cdot ...
4
votes
0answers
64 views
+200
When are nonintersecting finite degree field extensions linearly disjoint?
Let $F$ be a field, and let $K,L$ be finite degree field extensions of $F$ inside a common algebraic closure. Consider the following two properties:
(i) $K$ and $L$ are linearly disjoint over $F$: ...
2
votes
0answers
53 views
A basic question on factorization
Is the following true? If not, can anyone add some reasonable assumptions to make it true?
Statement. Let $E/F$ be a field extension and let $\alpha \in E \setminus F$ be algebraic over $F$. If $f(x) ...
1
vote
1answer
27 views
Field, Euclidean division question.
Let $K$ be a field and $f \in K[x]$. Show that if there is some $a \in K$ such that $f(a)=0$, then $x-a$ divides $f$.
My friend told me to use Euclidean division by $x-a$.
Also show that a ...
1
vote
0answers
29 views
A question regarding linear disjiontness and the degree of a field extension
Let $K / k$ and $L / k$ be field extensions with $K$ and $L$ linearly disjoint over $k$. Suppose that $K' / K$ is an extension of finite degree, and that all fields are characteristic zero.
Then is ...
0
votes
0answers
16 views
Help with proof concerning ordered fields and smallest elements
I have been trying to prove that every ordered field has no smallest positive element for a while now, and I think I have it worked out. However, I feel like something is missing in my proof. Here is ...
4
votes
4answers
79 views
To Show K is dense in $\mathbb{C}$
Let K be a subfield of $\mathbb{C}$ not contained in $\mathbb{R}$. Show that K is dense in $\mathbb{C}$.
completely stuck on it. can I get some help please.
0
votes
0answers
32 views
Transitive action on the set of algebra homomorphisms.
Let $k$ be a field, and $K/k$ be a Galois extension. Suppose $K'/k$ be an extension with $K'$ is a finitely generated $k$-algebra. Then the Galois group $\textrm{Gal}(K/k)$ acts canonically on the set ...
1
vote
1answer
51 views
Solve equations in a field with characteristic p.
Let $p$ be a prime and $K$ a field with characteristic $p$. How to solve the equation $x^2+2=0$ in the field $K$? Here $x, 2, 0 \in K$. Is it equivalent to solve the equation $x^2+2 = 0 \pmod p$? What ...
0
votes
1answer
82 views
A question regarding the finiteness of the degree of a field extension
Let $x$ and $y$ be transcendental and $y$ be algebraic over $\mathbb{Q}(x)$. Let $F$ be an algebraic Galois extension of $\mathbb{Q}(x)$ of infinite degree and let $\mathbb{Q}^{al}$ be the algebraic ...
1
vote
2answers
36 views
Question in Hungerford regarding field extensions
In Algebra by Hungerford, page 237 the sketch of proof for Theorem 1.10:
Theorem 1.10: If $K$ is a field and $f\in K[x]$ polynomial of degree $n$, then there exists a simple extension field $F = ...
-1
votes
4answers
53 views
Allowing the zero element in a field to have an inverse
In the definition of a field one of the required properties is that
every element other than zero has a multiplicative inverse.
It's vague whether the zero is forced not to have an inverse or not, ...
1
vote
2answers
36 views
Nonreal units in totally imaginary number fields
Suppose we are given a totally imaginary number field $L$ of degree greater $2$, is it possible that all units of $L$ lie in $\mathbb R$? This is true for almost all quadratic imaginary number fields, ...
3
votes
2answers
62 views
3 questions on field extensions
I am trying to figure out some things regarding field extensions and some questions have arisen on the way.
Let $a$ be a positive integer which doesn't have a rational $nth$ root:
Is the splitting ...
4
votes
3answers
102 views
Quadratic subfield of cyclotomic field
Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be ...
3
votes
1answer
48 views
Why don't I end up with the same splitting field?
I've understood that the splitting field of $x^4+2$ and the splitting field of $x^4-2$ over $\mathbb{Q}$ are both the field $\mathbb{Q}(\sqrt[4]{2} , i)$. With degree $8$ over $\mathbb{Q}$. This ...
6
votes
3answers
69 views
Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies
Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime.
Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
2
votes
1answer
39 views
Splitting field for $x^n+a$
What is a splitting field $E$ for $f(x)=x^n+a$ over the field $K$ of characteristic zero?
If I put $g(x)=x^{2n}-a^2=(x^n-a)(x^n+a)$. The splitting field $F$ of $g(x)$ is $K(\sqrt[n]{a},\alpha )$ ...
3
votes
1answer
39 views
Galois group of irreducible quartic with real coefficients
Let $K$ be a subfield of the real numbers and $f\in K[x]$ be an irreducible quartic. If $f$ has exactly two real roots, show that the Galois group of $f$ is $S_{4}$ or $D_{4}$ (I'm using the ...
3
votes
1answer
64 views
Irreducible polynomial over $\mathbb Z/7\mathbb Z$
Is the polynomial $t^6-3$ irreducible over $\mathbb{Z}/7\mathbb{Z}$? How would you prove that without doing all the computations? Is there a general method for this? Thank you.
1
vote
2answers
50 views
Example where $[E:\mathbb{Q}]<|\mathrm{Aut}_{\mathbb{Q}}E|$
Let $F$ be an algebraic closure of $\mathbb{Q}$ and let $E\subset F$ be a splitting field over $\mathbb{Q}$ of the set $\{x^{2}+a|a\in\mathbb{Q}\}$ so that $E$ is algebraic and Galois over ...
0
votes
1answer
27 views
Smallest Galois extension
Let $F$ be a finite dimensional Galois extension of $K$ and let $E$ be an intermediate field. Show that there is a unique smallest field $L$ such that $E\subset L\subset F$ and $L$ is Galois over $K$. ...
2
votes
1answer
32 views
Showing that a field extension is Galois
Let $F$ be an extension field of a field $K$. Let $E$ be an intermediate field such that $E$ is Galois over $K$, $F$ is Galois over $E$, and every $\sigma\in\mathrm{Aut}_{K}E$ is extendible to $F$. ...
8
votes
1answer
93 views
Algebraic closure of $(\mathbb{Z}/p\mathbb{Z})(T)$
Is there a concrete description of the algebraic closure of $(\mathbb{Z}/p\mathbb{Z})(T)$?
1
vote
1answer
31 views
Sum of the Hyperharmonic\Over-harmonic Series under $\mathbb{Z}_n$ for $p=2$
For $n \geq 5$ prime number, calculate the sum of:
$$1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{(n-1)^2}$$
under $\mathbb{Z}_n$.
I figured it's the hyperharmonic\over-harmonic series,
$$ ...
3
votes
5answers
57 views
On any finite field, adding the identity element a finite amount of times will result to the neutral element
Show that if you add the identity element ($1$) a finite amount of times will result to the neutral element ($0$).
I started with saying that in every field there's an element $a \in F$ so that $a + ...
0
votes
2answers
34 views
Addition table for a 4 elements field
Why is this addition table good,
\begin{matrix}
\boldsymbol{\textbf{}+} & \mathbf{0} & \boldsymbol{\textbf{}1} & \textbf{a} &\textbf{ b}\\
\boldsymbol{\textbf{}0} & 0 & 1 ...
3
votes
1answer
37 views
Lagrange interpolation of Galois field functions
I'm trying to understand how to apply Lagrangian interpolation for a function $F : GF(2^k) \to GF(2^k)$, but am having some difficulty. Suppose I have a function $F : GF(2^2) \to GF(2^2)$ given by the ...
3
votes
0answers
54 views
Hilbert’s zeros theorem, an application. (The algebraic variation)
Theorem: (Hilbert) If $k$ is a field, $A$ is a finitely generated $k$-algebra, and $M$ is a maximal ideal in $A$, then the factor $A/M$ is a finite extension of $k$. In particular if $k$ is ...
0
votes
0answers
24 views
If $K/L$ is normal and $L/F$ is purely inseparable, then $K/F$ is normal [duplicate]
This is a problem in Morandi's Field and Galois Theory, on page 49:
Let $F\subset L\subset K$ be field extensions such that $K/L$ is normal and $L/F$ is purely inseparable. Show that $K/F$ is ...
1
vote
1answer
49 views
Proving the order of a Galois group is equal to the dimension of $F$ over its fixed field w
Suppose $F/K$ is a finite dimensional field extension and $G = Aut_KF$. Let $G'$ be the fixed field of $F/K$, i.e. the set of members of $F$ which are fixed by every element of $G$. Before the ...
3
votes
1answer
46 views
Algebraically closed subfields of $\mathbb{C}$
What are the algebraically closed subfields of $\mathbb{C}$ ?
There is $\mathbb{C}$, there is $\bar{\mathbb{Q}}$... but what else ?
0
votes
0answers
21 views
An element of an extension field has a rational function with coefficients in the field that equals it
I stumbled across a statement without proof that I am having a hard time understanding or providing myself with intuition for. Any help would be most appreciated.
The statement reads:
Since $\xi$ is ...
0
votes
1answer
19 views
What does the adjoining mean?
What is the adjoin operation? The wikipedia link is pretty scant, but from eat it appears to be something along the lines as the smallest step towards the union of two sets? Are Adjunction and Union ...
0
votes
0answers
19 views
System of equations and Abel theorem
Consider this system of 3 equations to be solved in x,y and z:
$a x^m=(y+z)^n$
$by^m=(x+z)^n$
$cz^m=(x+y)^n$
The parameters $(a,b,c)$ and the unknown $(x,y,z)$ are all in $ℝ₊$. Also, m and n are ...
0
votes
0answers
26 views
Proof of Lempel-Golomb construction of Costas array
Can anyone please help me to prove Lempel-Golomb construction of Costas array, i.e., ${g_1}^i + {g_2}^j = 1$ forms costas array where $g_1$ and $g_2$ are primitive roots of a prime $p$ and $1\leq i ...
4
votes
4answers
182 views
Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$
I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
2
votes
2answers
74 views
No rational solutions of a system of equations
Please show that there does not exist $(a,b,c)\in\mathbb{Q}^3$ such that
\begin{matrix}
a^2b+2b^2c+2ac^2=0\\
a^2c+ab^2+2bc^2=0\\
a^3+2b^3+4c^3+12abc=3.
\end{matrix}
I'm able to show that this ...
1
vote
0answers
30 views
Showing $f(x)=g(x^{p^a})$ over field of (prime) characteristic $p>0$.
Let $f$ be a non-constant irreducible polynomial over a field $F$ of (prime) characteristic $p>0$. I need to prove that $f$ can be presented as:
$f(x)=g(x^{p^a})$, where $g$ is irreducible over ...
0
votes
0answers
26 views
Quadratic equation proof in field with characteristic field $\neq 2$
Suppose that I have a field $F$ with $char(F)\neq 2$. How one can prove quadratic formula in that field(?)?
Thank you.
8
votes
3answers
361 views
Does every algebraically closed field contain the field of complex numbers?
Does every algebraically closed field contain the field of complex numbers? Thank you very much.
5
votes
1answer
121 views
An exercise on cyclic extensions of Hungerford's book, Algebra.
I'm trying to show the following exercise of the book, it is in the section of cyclic extensions and says the following:
Let $\overline{\mathbb Q}$ be a fixed algebraic closure of $\mathbb Q$, let ...
1
vote
2answers
79 views
Roots of polynomials over finite fields
I've been trying to find the decomposition of $x^2-2$ to irreducible polynomials over $\mathbf{F}_5$ and $\mathbf{F}_7$.
I know that for some $a$ in $\mathbf{F}_5$ (for example), $x-a$ divides $x^2-2$ ...
0
votes
1answer
38 views
Polynomials decomposition into irreduceables
I've been trying to find the composition to irreduceables of the following polynomials with no much success:
X^2 +1 over the field F7
and X^2-2 over the field F5
Is there any method/algorithms I ...
1
vote
1answer
44 views
Field extension of $\mathbb Q$ of degree 2
Assume $K:\mathbb Q$ is a field extension and $[K:\mathbb Q] = 2$. Show that there is a unique squarefree $d \in \mathbb Z$ such that $K = \mathbb Q(\sqrt d)$.
I know that $K$ is generated by say ...
1
vote
1answer
40 views
the number of subfield $K$ of $L$ such that $\mathbb{Q}\subsetneq K\subsetneq L$
$\omega\neq 1 \in\mathbb{C}$ such that $\omega^3=1$, suppose $L$ be the field generated by $\omega, 2^{1/3}$ over $\mathbb{Q}$ i.e $L=\mathbb{Q}, (\omega,2^{1/3})$, the number of subfields $K$ of $L$ ...
1
vote
2answers
36 views
Dimension Recovery of $S \subset P_n(F)$
How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that
$f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset?
2
votes
2answers
46 views
Special elements of fields extensions
I was wondering if there is a method to find all elements $w\in F(\alpha_1,\ldots,\alpha_n)$ such that $F(w)=F(\alpha_1,\ldots,\alpha_n)$, where $\alpha_1,\ldots,\alpha_n$ are algebraic over the field ...
3
votes
3answers
122 views
Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.
I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
