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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5
votes
1answer
31 views

Generators of $\mathbb{F}_9/\mathbb{F}_3$ that do not generate $\mathbb{F}_9^{\times}$

Find a generator of the extension $\mathbb{F}_9/\mathbb{F}_3$ that does not generate the multiplicative group $\mathbb{F}_9^{\times}$. how many such elements exist? what are their minimal ...
0
votes
1answer
39 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
0
votes
3answers
83 views

showing $Q[\sqrt 2] = Q(\sqrt 2)$

The question came in my exam. $Q[\sqrt 2] = \{ a + b \sqrt2 \;| a,b \in Q\}$ and $Q(\sqrt 2)$ is minimal subfield of it's extension containing $Q$ and $\sqrt 2$. (In my book) It calls $F(a)$ ...
4
votes
1answer
41 views

Multiplicative order in field extension

Let $F/K$ be some field extension (both are finite fields) and $u$ be some element in $F$. I want to know if $u^{|K|} = u$ implies $u \in K$. And why?
0
votes
1answer
34 views

Proving that a function has no repeated roots

Let $\mathbb{Q} \subset F$ be a field extension. Prove that if $f(x) \in F[x]$ is irreducible, then it has no repeated roots in any field extension of F. as a hint we were given that a repeated root ...
0
votes
0answers
38 views

Notational Clarification - Abstract Algebra

I'm going through a paper on homomorphic encryption by Smart and Vercauteren entitled "Fully Homomorphic SIMD operations" and had a question about some notation used in the paper. In section 2 of the ...
5
votes
1answer
78 views

Difficult algebraic problem - irreducible polynomials

Let $p$ be a prime number and $q=p^n$ with $n \in \mathbb{N}$. We define the polynomial$$ F_m:=\prod \left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, nomic}, \text{deg}(f)=m \right\rbrace $$ ...
1
vote
3answers
45 views

$\mathbb Q[x]/\langle x^2+1\rangle$ is a splitting field of $x^2+1$ over $\mathbb Q.$

Due to the Kronecker's theorem, $x^2+1\in\mathbb Q[x]$ splits over $\mathbb Q[x]/\langle x^2+1\rangle.$ But how to show that $\mathbb Q[x]/\langle x^2+1\rangle$ is a splitting field of $x^2+1$ over ...
2
votes
2answers
36 views

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5. I will just add that this task is slightly ahead of my knowledge of field theory. So any pointers would be ...
0
votes
0answers
13 views

How many basis are in n-dimensional vector space over field of q-elements

How many basis are in n-dimensional vector space over field of q-elements? Thanks in advance! Any help is appreciated
0
votes
1answer
18 views

Why does the 2-D vector space would imply the splitting field?

Why does the 2-D vector space would imply the splitting field?
0
votes
2answers
33 views

Shouldn't the induction hypothesis be taken only on $n?$

I'm having problem in getting the underlined statement from Gallian text: Shouldn't the induction hypothesis be taken only on $n?$ But here the author also assumed the case for arbitrary field in ...
0
votes
1answer
21 views

How to show that $\mathbb Q[\sqrt 2]$ is the smallest subfield of $\mathbb R$ containing $\mathbb Q$ and $\sqrt 2?$

How to show that $\mathbb Q[\sqrt 2]=\{a+b\sqrt 2:a,b\in\mathbb Q\}$ is the smallest subfield of $\mathbb R$ containing $\mathbb Q$ and $\sqrt 2?$ I've shown that $\mathbb Q[\sqrt 2]$ is a subfield ...
0
votes
1answer
49 views

Order of element of multiplicative group of finite field mod polynomial

If $K$ is a finite field of size $q$ and $f$ is a degree $n$ polynomial in $K[x]$, then we can form the quotient field by modding out this polynomial. Elements of this quotient field are of the form ...
0
votes
1answer
21 views

Is it jusfied then to say $\mathbb R$ the splitting field of $x^2-1$ over $\mathbb R?$

Gallian text define splitting field as: Take for example $x^2-1\in\mathbb R[x]$ Due to the definition $x^2-1$ splits over $\mathbb R$ as well as over $\mathbb Q,$ a proper subfield of $\mathbb R.$ ...
3
votes
1answer
33 views

Simplifying presentation of elements of finite field

Let me describe my question through an example. Finite field of order (for example) 8 can be constructed as $\mathbb{F}_8 = \mathbb{F}_2[t]/(t^3 + t + 1)$. So one of a natural presentation of the ...
2
votes
3answers
19 views

Shouldn't the yellow marked $a_0$ be $a_0+\langle p(x)\rangle?$

I'm having problem in getting the proof from Gallian text in the higlighted region: Shouldn't the yellow marked $a_0$ be $a_0+\langle p(x)\rangle?$ Edited: Shouldn't the $a_i$'s in the ...
0
votes
0answers
19 views

Notational issue

Let $K = F(t)$. If $r \in K: (\nexists c \in F: r(t) = c \forall t)$ is a rational function and $L = F(r(t))$, then what form does $f \in L$ have? Is it a rational function where the coefficients are ...
0
votes
1answer
33 views

Concerning a Cyclic Galois Group

Why is it that: $\forall K \supseteq \mathbb{Q}(\mathbb{i}), G=Gal(K / \mathbb{Q}) = \langle \sigma \rangle \implies \sigma (\mathbb{i}) = - \mathbb{i}$? (Note: I am guessing that $\sigma ≠ ...
0
votes
1answer
26 views

$\mathbb R[x]/\langle x^2+1\rangle\simeq\mathbb C$ and $\mathbb R[x]/\langle x^2+1\rangle$ contains a zero of $x^2+1$

How to show that $\mathbb R[x]/\langle x^2+1\rangle\simeq\mathbb C$ and $\mathbb R[x]/\langle x^2+1\rangle$ contains a zero of $x^2+1$ Due to the division algorithm, $\mathbb R[x]/\langle ...
2
votes
2answers
40 views

Show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois; prove that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}(2^{1/2})$ is Galois

I would like to show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois. Can I just say that it is not separable because $2^{1/4} \in \mathbb{Q}(2^{1/4})$ but its minimal polynomial in $\mathbb{Q}$ ...
-1
votes
1answer
15 views

Permutation of a fixed field is an intermediate field corresponding with the conjugate of the group corresponding to the fixed field

The following is my question: Let $K/F$ be a Galois extension with Galois group $G = Gal(K/F)$, with intermediate field $L: F \subseteq L \subseteq K$ which corresponds to subgroup $H \leq G$ by the ...
1
vote
1answer
29 views

Every Intermediate Field of Abelian Galois Field Extension is Splitting Field of a Separable Polynomial

This is my question: Suppose the $K/F$ is a Galois extension with an abelian Galois group $G$. Prove that every intermediate field $L: F \subseteq L \subseteq K$ is the splitting field (over $F$) of ...
1
vote
0answers
31 views

Union of field extensions over Q

I am asked to prove that $L=\bigcup_{n=1}^\infty\mathbb{Q}(\sqrt[n]2)$ is an algebraic field extension over $\mathbb{Q}$. So far I have: Let $\beta\in L$, then by definition of union there exists a ...
1
vote
2answers
39 views

Prove the ring $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ has inverse and is a field

How can I prove that $\frac{1}{a+b\sqrt[3]{2}+c\sqrt[3]{4}}$ is of the form $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ (i.e. that $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ is a field) for all rational $a,b,c$ and ...
3
votes
1answer
47 views

Algorithm to find representation of an element of field extension $\mathbb{Q}(q)$ in the form $\sum a_i q^i$

Let $\mathbb{Q}(q)$ be a field extension of $\mathbb{Q}$, where $q$ is a real root of some monic irreducible polynomial $p(x) \in \mathbb{Z}[x]$ of degree $d=3$. Given $x \in \mathbb{R}$, (or ...
2
votes
1answer
79 views

Why characters are continuous

According to Wikipedia: ''Every character is automatically continuous from $A$  to $\mathbb C$, since the kernel of a character is a maximal ideal, which is closed. '' where $A$ is a Banach algebra. ...
0
votes
0answers
30 views

Need some help finishing this proof about characters in Banach algebras

I tried to prove: Let $A$ be a commutative unital complex Banach algebra. Then there is a bijection between the maximal ideals in $A$ and the set of non-zero homomorphisms $A \to \mathbb C$. But I ...
3
votes
1answer
40 views

Number of isomorphisms between two fields

Let $F,F'$ be two fields. Is there anything that can be said about the number of isomorphisms that can exist? In particular can there be more than one? What if $F$ is the complex numbers $\mathbb C$? ...
1
vote
2answers
70 views

When does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$?

Let $p$ be a prime integer, and let $q=p^r$ and $q'=p^k$. For which values of $r$ and $k$ does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$? From Artin's Algebra, Chapter 15, problem 7.12 from the ...
2
votes
1answer
65 views

Why $K(u)$ is a field?

Let $F$ be an extension field of $K$ and $u\in F$. How do we know that adjoining an element of F to K, makes $K(u)$ a field? I know that $Q(\sqrt2)=\{a+b\sqrt2|a,b\in Q\}$ is a field, but in the ...
0
votes
3answers
59 views

Proof about finite dimensional vector spaces over fields

Prove that every finite dimensional vector space $V $of dimension $n$ over a field $F$ is isomorphic to the vector space $F^n$. Okay, lot's of stuff here. I think most of the reason I can not do this ...
2
votes
2answers
41 views

Ring homomorphism with field as image, is the pre-image also a field?

Let $f:R\rightarrow S$ be a surjective ring homomorphism. $R,S$ are both integral domains. Suppose $S$ is a field, then is $R$ also a field? A possible useful fact: A finite integral domain is a ...
1
vote
2answers
36 views

Analogy between the quaternion ring and extensions of the rationals

I've started studing fields and their extensions. As an exercise I proved that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$ by showing that $B=(1,\sqrt2,\sqrt3,\sqrt6)$ is a base for the extension field ...
3
votes
1answer
71 views

Find primitive element of splitting field of $1 + x + x^2 - x^5$

As the title says, I need to find the primitive element of the splitting field of $1 + x + x^2 - x^5$ over $\mathbb{Q}$. Firstly, I would proceed by finding the roots as the splitting field has to ...
0
votes
3answers
29 views

Ring theory question.

Why is a field with 27 elements has characteristic 3? I was solving a question and I came to know this fact which I didn't know before. Is there anyone who can explain this to me? Thanks in advance. ...
1
vote
2answers
40 views

Constructing expressions

Suppose you are given all the elementary symmetric functions of $n$ variables $x_1,x_2,...,x_n$ and two rational functions $A(x_1,x_2,...,x_n)$ and $B(x_1,x_2,...,x_n)$ in the same $n$ variables that ...
2
votes
2answers
79 views

Field extension of a finite field

Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements. I am not sure how to do this ...
4
votes
2answers
76 views

Group-Isomorphism problem

I want to find an group-isomorphism $$ \psi : (\mathbb{Z}/8\mathbb{Z},+) \longrightarrow \mathbb{F}_9^\times $$ which should be used to multiply elements in $\mathbb{F}_9$ or to find the inverse ...
0
votes
1answer
33 views

Extension Fields and Quotients

In the Dummit and Foote 3ed chapter on field extensions (ch. 13), it is stated as a theorem (6) that $ F(\alpha) \cong F[x]/(p(x))$ where $\alpha$ is a root of $p(x)$ and goes on to state that any ...
3
votes
2answers
96 views

Field extensions and algebraic/transcendental elements

Let $E$ be an extension of field $F$, and let $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over $F$ but algebraic over $F(\beta)$. Show that $\beta$ is algebraic over $F(\alpha)$. ...
1
vote
1answer
33 views

Splitting field in $\mathbb{C}$ over $\mathbb{Q}$

I want to find the splitting field in$\mathbb{C}$ of $x^4-4$ over $\mathbb{Q}$. I solved for the zeros, which is $i\sqrt2, -i\sqrt2, \sqrt2, -\sqrt2$, so the splitting field, say $E$, is just ...
8
votes
2answers
162 views

$\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}$?

Is there an easy way to see that $$\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}?$$ I know that $\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})$ is a subfield of ...
1
vote
1answer
43 views

Computing a field extension by hand

Let $k$ be a field, and $K=k(t)$ be the field of rational functions (where $t$ is indeterminate). Let $F=k(t^2)$. A typical element of $F$ will look like: $$ \frac{\displaystyle \sum_{i=0}^{n} a_{2i} ...
5
votes
0answers
56 views

The reducibility of polynomial $x^{2^n}+x+1$ over $Z_2$ [duplicate]

I meet the problem in Hungerford's Algebra, page 282 as follows: If $n>2$, then the polynomial $x^{2^n}+x+1$ is reducible over $Z_2$ . I have no any idea on this. But I really want to know how to ...
1
vote
1answer
34 views

Splitting field in multiple field extension

Let $F\subset K\subset E$ be fields, $f(x)\in F[x]$. E is the splitting field of $f(x)$ over $F$. Prove $E$ is the splitting field of $f(x)$ over $K$. My problem with this proof is that I do not know ...
5
votes
0answers
86 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
1
vote
1answer
61 views

Splitting field of a set of separable polynomials implies separability of extension.

Let $F$ be a splitting field of $S\subset K[x]$ over $K$, where $S$ is a set of separable polynomials. I want to show that $F$ is separable over $K$, meaning for all $u\in F-K$, the irreducible ...
0
votes
0answers
17 views

Application of the Isomorphism Extension Thm

Let $K$ be an algebraic closed field. Show that every isomorphism $\sigma$ of $K$ onto a subfield of itself $s.t. \ K$ is alg. $/ \sigma[K]$ is an automorphism of $K$? this seems too clear to me that ...
0
votes
2answers
49 views

Prove: 2^(1/3) cannot be written in terms of any given root of unity

How does one prove that there exists no natural number $n$ such that $\sqrt[3]{2} $ belongs to the field extension of the rationals by the $n$th root of unity?