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 topic....

learn more… | top users | synonyms (1)

1
vote
1answer
76 views

How does one construct the Galois field extension $GF((2^2)^3)$?

Looking at past exam question, one asks us to construct a Galois field extension $GF((2^2)^3)$ whenever the primitive irreducible polynomial $p(X) = X^3 + \alpha X^2 + \alpha X + \alpha \in GF(2^2)[X]$...
2
votes
1answer
44 views

Finding the subfields of the cyclotomic field of order $5$

This is part of an exercise from Hungerford's Algebra: Find all intermediate fields in the field extension $F_5/\mathbb{Q}$, where $F_5$ is the cyclotomic extension of $\mathbb{Q}$ of order $5$. ...
1
vote
1answer
22 views

Abstract algebra. Proof of: Let $F$ be a finite field and $P$ an irreducible polynomial upon $F$. Then $(F[t]|_{\equiv_P}, + , \ast)$ is a field.

Division of polynomials I put what is unclear to me in between three asterisks bounding the unclear lines... $\equiv_{P}$ is defined as ($\forall Q, S \in F[t]$) $Q\equiv_{P} S \iff P|Q-S$ ...
0
votes
0answers
35 views

Prove that the coefficients of a polynomial are in a finite field

I am trying to understand the proof of the following statement: Let $\mathscr{θ}$ be an algebraic element over the finite field $F$ and $\mathscr{θ=θ_1,θ_2 ... θ_n}$ be all the conjugate elementes of ...
2
votes
1answer
110 views

Ring of integers of a cyclotomic number field

Let $\omega$ be the primitive $n^{th}$ root of unity. Consider the number field $\mathbb Q(\omega)$. How to show that the ring of integers for this field is $\mathbb Z(\omega)$? Also, find the ...
5
votes
0answers
82 views

How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$?

$\mathbb{F}/\mathbb{Q}$ is a finite extension. Denote $\dim_{\mathbb{Q}}\mathbb{F}=n$. How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$? Thoughts: if $A=\{v_1,...,v_n\}...
2
votes
2answers
46 views

Is it true that for every $n\in\mathbb{N}$ there exists an algebraic curve $C$ and a point $p$ in that curve whose tangent plane has dimension $n$?

The title is very self-explanatory, I was thinking of finding a curve $k^n$ (k is the field with a non-singular point p, such that its tangent plane is V(0) and thus would be $k^{n}$ whose dimension ...
2
votes
2answers
56 views

Naïve groups, fields and ideals

Please excuse the simplicity of this question, but I am very new to groups and fields. I only seek an simplistic / intuitive expalnation, and confirmation / refutation re whether I am on the right ...
4
votes
2answers
100 views

Why is this Galois group abelian?

Consider the field extension $\mathbb Q(\zeta_3,\sqrt[3]2,\zeta_8)/\mathbb Q(\zeta_3)$, with intermediate fields $\mathbb Q(\zeta_3,\sqrt[3]2)$ and $\mathbb Q(\zeta_3,\zeta_8)$. Denote $L:=\...
2
votes
1answer
57 views

Reducing splitting field

If we have splitting field: $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{2-i\sqrt{6}}) $$ we can multiply these two zeroes and get $\sqrt{10}$ so we have $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{10})$$ ...
1
vote
2answers
53 views

Are the fields $\mathbb{Q}(\sqrt[7]{16}+3 \sqrt[7]{8})$ and $\mathbb{Q}(\sqrt[7]{16})$ equal?

I have trouble with these field extensions. Is field $\mathbb{Q}(\sqrt[7]{16}+3 \sqrt[7]{8})$ equal to field $\mathbb{Q}(\sqrt[7]{16})$? We can $\sqrt[7]{16}+3 \sqrt[7]{8}$ express as $(\sqrt[7]{2}+...
2
votes
1answer
73 views

Cyclic extension without primitive root of unity

Let $F$ be a field that doesn't contain a primitive fourth root of unity. Let $L = F(\sqrt a)$ for some $a \in F - F^2$ and let $K=L(\sqrt b)$ for some $b \in L - L^2$. If we have $N_{L/F}(b) \...
1
vote
3answers
59 views

List the elements of the field $K = \mathbb{Z}_2[x]/f(x)$ where $f(x)=x^5+x^4+1$ and is irreducible

Since $\dim_{\mathbb{Z}_2} K = \deg f(x)=5$, $K$ has $2^5=32$ elements. So constructing the field $K$, I get: \begin{array}{|c|c|c|} \hline \text{polynomial} & \text{power of $x$} & \text{...
1
vote
1answer
79 views

Unique isomorphisms and universal properties

Having not studied category theory, I'm trying to piece together without using category theory what is meant when an algebraic structure is said to possess a universal property or be unique up to ...
4
votes
2answers
91 views

Determine which of the following rings are fields.

Have I done it correctly? Determine which of the following rings are fields: a) $(\mathbb{Z}/2\mathbb{Z})[x]$/$\large_{(x^2+1)}$ b)$(\mathbb{Z}/3\mathbb{Z})[x]$/$\large_{(x^2+1)}$ My ...
3
votes
1answer
69 views

What are the groups $\text{Hom}(F^\times\!, F^+\!)$ and $\text{Hom}(F^+\!, F^\times\!)$?

Background. Exercise 36 in Rose's A Course On Group Theory reads Prove that there is no field $F$ with $F^\times \cong F^+$. The problem is solved in characteristic $\ne 2$ by considering $-1$ ...
3
votes
1answer
116 views

Proof that every field $F$ has an algebraic closure $\bar F$

I am reading the book A First Course in Abstract Algebra written by Fraleigh and I do not really understand the proof of theorem 31.22, that every field $F$ has and algebraic closure $\bar F$. I ...
1
vote
0answers
41 views

How to solve the equation $x^2+Dy^2=\alpha$ over finite fields

It is known that the equation $x^2+Dy^2=1$ is solved over finite fields $\mathbb{F}_q$ and we can point out the solutions . I wonder can we give solutions for the equation $x^2+Dy^2=\alpha$ for any $\...
1
vote
1answer
152 views

Question from 14.6 “Galois Groups of Polynomials” from Dummit and Foote

I am confused in the proof of proposition 30 in Dummit and Foote on page 608. Near the end of this "proof" he goes on to say, By the Fundamental Theorem of Galois Theory, the fixed field of $S_{n}...
3
votes
2answers
62 views

How can we work with field extensions when our base fields aren't actually subfields?

I've been wondering this for a little while. Say we are working with the rational numbers $\mathbb{Q}$, and then we wish to talk about the extension fields $E$ of $\mathbb{Q}$, by which we mean the ...
0
votes
1answer
89 views

Galois group for the algebraic closure of a finite field

If $N$ is the algebraic closure of a finite field $F$, prove that $\operatorname{Gal}(N/F)$ is an abelian group and that any element of the Galois group has infinite order. If $N$ were a finite ...
0
votes
1answer
16 views

Splitting of primes terminology doubt

What do we mean when we say that a given prime $p$ splits completely in an algebraic extension of $\mathbb Q$? Are we talking about the splitting of prime ideals into unique factors? And, in that ...
2
votes
2answers
54 views

How should I find Splitting Field of $x^3-2$ over $\mathbb Q$.

How should I find Splitting Field of $x^3-2$ over $\mathbb Q$. **My try **: $x^3-2=(x-2^\frac{1}{3})(x^2+2^\frac{1}{3}x+2^\frac{2}{3})$ On solving I am getting the roots as $2^\frac{1}{3},\dfrac{2^...
1
vote
1answer
41 views

Field Question Proofs

True or False: In every field $F$, if $x,y$ belong to $F$ and $w,w'$ belong to $F$ such that $x * w = 1$ and $y * w' = 1$, then $(x * y) * (w * w') = 1$. I think the answer would be false mainly ...
0
votes
0answers
34 views

Galois closure uniqueness confusion

I'm a little bit confused by the statement of this corollary, (Corollary 23 pg 594 of Dummit and Foote) Let $E/F$ be any finite separable extension. Then $E$ is contained in an extension $K$ which is ...
2
votes
1answer
76 views

From where can I study more about Dickson polynomials?

I know some basic bits about this construction as to how they effect permutations of Galois fields. But I want to get some detailed understanding of them. Any references?
2
votes
1answer
61 views

Field extension of prime degree

Question: Let $L$ be the extension of the field $K$ such that $[L:K]=p$, where $p$ is a prime number, and $\alpha \in L$. Prove that $K(\alpha)=K$ or $K(\alpha)=L.$ Proof: From $$ \alpha \in L \...
1
vote
0answers
52 views

Find and show primitive element of field extension

I have the polynomial $\ f(x)= x^{15}-3 $ and want to find the splitting field. The splitting field is surely $\Bbb Q$ adjoined the roots of the polynomial. I want to write the splitting field as $\...
1
vote
0answers
44 views

What does this theorem mean, exactly?

The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible polynomials in $F_p[X]$ of degree $d$ where $d$ runs through all the divisors of $n$. I don't even get the ...
7
votes
1answer
83 views

If there are $k_1, k_2 \in K$ such that $K(\alpha + k_1\beta)=K(\alpha + k_2\beta)$ then $K(\alpha,\beta) = K(\alpha + c\beta)$ for some $c \in K$.

Here is the problem: "Let $K \subset M$ be a finite field extension, and $\alpha, \beta \in M$. Suppose there are $k_1, k_2 \in K$ are distinct and such that $K(\alpha + k_1\beta)=K(\alpha + k_2\beta)$...
-1
votes
1answer
62 views

Cyclotomic field over $\Bbb Q$

Let $K$ be cyclotomic field generated over $\Bbb Q$ by the $9$th root of unity $z$, having Galois group $G$. Show that it is a cyclic extension of degree $6$ of $\Bbb Q$ and by making use of the ...
3
votes
3answers
118 views

$\alpha \in \overline{\mathbb{F}}_q$ satisfying $\alpha^{q+1}+\alpha=-1$

Let $\overline{\mathbb{F}}_q$ be the algebraic closure of $\mathbb{F}_q$. Assume that $\alpha \in \overline{\mathbb{F}}_q$ satisfies at $$\alpha^{q+1}+\alpha=-1$$ Show that $\alpha \in \mathbb{F}_{q^...
1
vote
1answer
35 views

Finding proper subfields

Let $\omega$ denote the cube root of unity such that $\omega\neq 1$. I want to find the subfields properly contained in $\mathbb Q(\sqrt[3]{2},\omega)$ and containing $\mathbb Q$ properly. Two of ...
0
votes
1answer
39 views

Relation between $\mathbb{Q}(2^{1/3})$ and $\mathbb{Q}(w2^{1/3})$

I have the following exercise in my homework: Are $\mathbb{Q}(2^{1/3})$ and $\mathbb{Q}(w2^{1/3})$ isomorphic, where $w = \textrm{cis}((2\pi)/3)$? Prove your answer. I think they are, but I'm ...
2
votes
5answers
78 views

looking for the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$ with $\alpha=\sqrt[3]{2}$

I have $\alpha=\sqrt[3]{2}$ and want to calculate the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$. There's a hint which tells me to look at the minimal polynomial $m_\alpha$ of $\alpha$ over $\Bbb ...
0
votes
1answer
32 views

Is the statement of the theorem correct?

I have been asked to prove this:: $f,g$ are polynomials over a field $F$ .Prove that if $f,g$ are relatively prime then $f,g$ have no common roots in any extension of $F$. But I wonder why is ...
4
votes
2answers
23 views

Non-algebraic subfield intersection

Construct an example in which a field $F$ is of degree 2 over two distinct subfields $A$ and $B$, but so that $F$ is not algebraic over $A\cap B$. I'm having trouble thinking of an explicit example ...
5
votes
1answer
162 views

Showing that $\mathbb{Q}(\sqrt{2},\sqrt{3}, \sqrt{(9 - 5\sqrt{3})(2-\sqrt{2})})$ is normal over $\mathbb{Q}$, and finding its Galois group

If $K=\mathbb{Q}(\sqrt{2},\sqrt{3}, u)$, where $u^2 = (9 - 5\sqrt{3})(2-\sqrt{2})$, show that $K/\mathbb{Q}$ is normal, and find $\operatorname{Gal}(K/\mathbb{Q})$. I found that the minimal ...
0
votes
1answer
92 views

finding intermediate fields of field extensions: simple method vs. Galois theory

I am looking for a straightforward way to find all intermediate fields of a field extension. Let's take the splitting field of $X^3-2$ over $\Bbb Q$ as an example. If we adjoin the roots of $X^3-2$ ...
1
vote
1answer
67 views

How do I find the quotient field of $\mathbb{Z}[\sqrt{d}]$?

Our teacher said sometimes the quotient field is $\mathbb{Q}[\sqrt{d}]$ and sometimes it's $\mathbb{Q}[\frac{1+\sqrt{d}}{2}]$. How do we decide, or what are the conditions on $d$ which helps us to ...
2
votes
5answers
81 views

Basis of $\mathbb{Q}[\sqrt[3]{2}]$

How do I prove that $1, \sqrt[3]{2}, (\sqrt[3]{2})^2$ is a basis of $\mathbb{Q}[\sqrt[3]{2}] = \{ a + b \sqrt[3]{2} + c (\sqrt[3]{2})^2\: a,b,c \in \mathbb{Q} \}$. It's one of these cases where the ...
8
votes
1answer
85 views

On the existence of field morphisms

Let $K$ and $L$ be two fields, does the existence of two field morphisms $f\colon K\rightarrow L,\ g\colon L\rightarrow K$ imply that, as abstract fields, $K\cong L$ (not necessarily via $f$ or $g$)?
0
votes
1answer
35 views

Exactly one ring homomorphism $F[X] \rightarrow S$

Let $F$ be a field, and $f \in F[X]/(f)$. Let $f$ have a zero point $\alpha$, that is, $f(\alpha)=0$. Let $F$ be a subring of $S$, and $\beta \in S$ with $f(\beta)=0$. Show that there is exactly one ...
0
votes
2answers
34 views

Let $K$ be a field and let $p(x)\in K[x]$ be an irreducible polynomial of degree $d$. Let $L = K[x]/p(x)$. Prove that $[L:K] = d$.

I'm not sure where to go with this question. I know that $K[x]/p(x)$ is a field since p$(x)$ is irreducible means it is maximal in $K[x]$.
2
votes
3answers
323 views

How do I prove this field homomorphism is an isomorphism?

The question is as follows. Let $F$ be a finite field with unit $1$ not equal to zero. Let the function $f: F \to F$ be given by $f(x) = x^3$, where the $\operatorname{char}(F) = 3$. Prove it is a ...
5
votes
1answer
51 views

Is the primitve element of $\mathbb{Q}[\alpha_1, \alpha_2, \ldots]/\mathbb{Q}$ always $\alpha_1 + \alpha_2 + \cdots$?

I have dealt with a number of algebraic field extensions $\mathbb{Q}[\alpha_1, \alpha_2, \ldots]/\mathbb{Q}$ and the primitive element was always $\alpha_1 + \alpha_2 + \cdots$. Is this generally true ...
1
vote
2answers
46 views

Injectivity and norm function on finite fields [closed]

Let $q$ be an odd prime power. Consider the map $f:\Bbb F_{q^3} \rightarrow \Bbb F_{q^3}$, defined by $$f(x)=\alpha x^q+\alpha^q x$$ for some fixed $\alpha \in \Bbb F_{q^3} \setminus \{ 0 \}$. ...
2
votes
1answer
22 views

A basis of a field extension $K \subset L$ spans $L(C)$ over $K(C)$ for any subset $C$ of a field $M\supset L$

Let $K \subset L \subset M$ be fields; $\{\beta_1, ..., \beta_k\}$ a basis for $L$ over $K$ and $C$ a subset of $M$. Then $\{\beta_1, ..., \beta_k\}$ generates $L(C)$ over $K(C)$ (where $K(C)$ is the ...
6
votes
3answers
119 views

A commutative noetherian ring in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields

PROBLEM A commutative noetherian ring $R$ in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields. I am lost with the condition $I^2=I$ and the desired result "a finite ...
1
vote
0answers
79 views

Extensions of a field?

Prove that there are infinitely many degree 5 extensions of $\mathbb{F}_{121}(x)$. I know that $\mathbb{F}_{121}$ is isomorphic to the splitting field of $x^{121}-x$ over $\mathbb{F}_{11}$, but I'm ...