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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8
votes
4answers
268 views

Can we turn $\mathbb{R}^n$ into a field by changing the multiplication?

Of course $\mathbb{R}$ is a field with usual addition and multiplication. When we move up a dimension into $\mathbb{R}^2$, however, there is not a clear way to multiply two vectors together to get ...
1
vote
1answer
69 views

An algebraically closed field with characteristic $p>0$

I want to know about an algebraically closed field that is not of characteristic $0$. I really don't know about infinite fields with characteristic $p$ so I will appreciate your comments.
3
votes
1answer
121 views

Intermediate fields between $\mathbb{Z}_2 (\sqrt{x},\sqrt{y})$ and $\mathbb{Z}_2 (x,y)$

Let $K=\mathbb{Z}_2 (x,y)$, where $x,y$ are independent, and $L$ be a splitting field extension of $(X^2 - x) (X^2 - y)$, then $[L:K] = 4$ and $L = K(\sqrt{x},\sqrt{y})$ where $\sqrt{x},\sqrt{y}$ are ...
4
votes
2answers
116 views

What is the field of definition of an invariant ideal?

Let $K/k$ be a finitely generated field extension, such that $k=K^G$ for some (possibly infinite) set $G$ of automorphisms of $K$. Now, consider the extension of polynomial rings $$ ...
2
votes
0answers
127 views

Another galois theory problem. (quartic with dihedral Galois group)

Given: If $f$ in $k[X]$ is an irreducible quartic and $G$ is the galois group of its splitting field, then $G$ is contained in $D_8$ iff the resolvent cubic of $f$ has a root in $k$. Now suppose ...
1
vote
1answer
125 views

Which of the following statement is not necessarily true for the product of rings $R \times R$ when it is true for $R$?

$R$ is a ring. Which of the following statements is not necessarily true for the product of rings $R \times R$ when it is true for $R$? A. There exists some generator whose order is finite. B. $R$ ...
1
vote
1answer
49 views

Showing $F$ of Characteristic $p$ is Separable Provided $\phi(a) = a^p$ is Surjective on $F$

Let $F$ be a field of characteristic prime $p$. Let $\phi: F \rightarrow F$ be defined as $\phi(a) = a^p$ for all $a \in F$. Goals: (i) Show that $\phi$ is an injective homomorphism of $F$. (ii) ...
4
votes
1answer
353 views

Number of intermediate field of a finite separable extension

Exercise A-30 in Milne's Fields and Galois Theory notes is: Let $L/K$ be a separable algebraic extension of degree $d$. Show that the number of fields between $K$ and $L$ is at most $2^{d!}$. ...
0
votes
3answers
106 views

divisibility question in abstract algebra over a field

$d|n \Rightarrow x^{p^d} - x$ divides $x^{p^n} - x$ over $\mathbb{F}_{p}$ where $\mathbb{F}$ is a field. Attempt: $d|n \Rightarrow x^{p^d} - x$ divides $x^{p^n} - x \Rightarrow x^{p^d - 1} - 1$ ...
2
votes
2answers
54 views

In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$.

In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$. I know how to find the inverses of elements within sets, rings, and fields. I know what to do if the field ...
1
vote
2answers
105 views

The polynomial $p(x)=x^4+x+1$ can be shown to be irreducible over $\mathbb{Z}_7$. Show that $\mathbb{Z}_7[x]/\langle p(x)\rangle$ is a field.

The polynomial $p(x)=x^4+x+1$ can be shown to be irreducible over $\mathbb{Z}_7$. Show that $\mathbb{Z}_7[x]/\langle p(x)\rangle$ is a field. Since $p(x)$ is irreducible over $\mathbb{Z}_7$, then ...
6
votes
2answers
338 views

Centre of a simple algebra is a field

How can one show that the centre of simple algebra is a field? I have tried it and proved that the inverse exists for every element of centre but cannot prove that inverse of every element ...
1
vote
2answers
67 views

polynomial with integer coefficients

Question: Let $\Pi_{j=1}^n (z-z_j)$ be a polynomial with integer coefficients. Is also $\Pi_{j=1}^n (z-z_j^k)$ for $k=1,2,3,\dots$ a polynomial with integer coefficients? In fact, this is a question ...
2
votes
4answers
324 views

Prove or disprove that $\mathbb{Z}[x] / (x^3 - 2)$ is a field.

Prove or disprove that $\mathbb{Z}[x] / (x^3 - 2)$ is a field. I am pretty sure it is not a field because if you consider $F = \mathbb{Z}[x] / (x^3 - 2)\mathbb{Z}[x]$ where $F$ is a field, then char ...
1
vote
0answers
82 views

What is the degree of $\sqrt[5]{16}$ over Q?

The question is in the title. Obviously I am not looking for the actual degree, just some tips on how to find it. Eisenstein's criteria doesn't immediately apply to the polynomial $x^5-16$ so I ...
3
votes
2answers
96 views

Is $i$ contained in this field extension?

As part of a larger problem I need to show that $i$ is not contained in the field extension $\mathbb Q(\sqrt[3]{2},\zeta)$, where $\zeta$ is the third root of unity. I understand that the ...
4
votes
1answer
127 views

Questions about field extensions and irreducibility

These questions are related to Galois theory, but are general field theory questions. First question: I understand Eisenstein's criterion for proving irreducibility of polynomials over $Q$. ...
0
votes
1answer
77 views

Showing Irreducible $f(x^{p^m}) \in F[x]$ is Separable if $Char(F) = p$

Let $F$ be of characteristic $p \in \mathbb{N}$ and $f(x) \in F[x]$ be an irreducible polynomial s.t. $f(x) = \sum_i a_i x^{n_i} $ where for each $i$, $a_i = 0$ or $p\,\,|\,\, n_i$. That is, we can ...
2
votes
2answers
163 views

If a commutative ring with identity is the sum of two ideals, then their product is equal to their intersection.

My problem is to prove exactly as the title says; particularly if I+J=R for some commutative ring R with identity and ideals I and J of the ring R, then IJ = I ∩ J. I know already that IJ is an ideal ...
2
votes
1answer
130 views

Galois theory problem.

Let $K$ and $L$ be Galois extensions of $k$, contained in an extension $M$ of $k$ ( so that $KL$ makes sense). Show that if $K\cap L=k$, then $\operatorname{Gal}(KL/k)\cong ...
2
votes
0answers
70 views

On cyclic extensions.

I would really appreciate it if you help me with the following problem: Suppose K is algebraically closed, f is an automorphism of K of in finite order, and k is the elements of K fixed by f. Show ...
1
vote
1answer
225 views

Uniqueness of algebraic closure of $\Bbb{Q}$ in $\Bbb{C}$.

I ask 2 questions. 1) How can I prove the following sentence? If $\Bbb{Q}'$ is the subset of $\Bbb{C}$ consisting of all algebraic elements over $\Bbb{Q}$, then $\Bbb{Q}'$ is an algebraic closure of ...
7
votes
2answers
592 views

How does $\cos(2\pi/257)$ look like in real radicals?

We know $\cos(2\pi/p)$ for p a Fermat prime can be expressed in real radicals. The case $p=17$ is a root of an 8th deg eqn, but can be also given as a sequence of nested radicals, $$\begin{aligned} ...
1
vote
0answers
66 views

Primitive Element Theorem

I am trying to understand the proof of the primtive element theorem, as written in Milne's notes. It is not emphasized there where the assumption that $\beta$ is separable over $F$ is used in the ...
0
votes
1answer
32 views

Are coefficients and values for x in F[x] in the same set?

I'm trying to understand the construction of $F[x]$ where $F$ is a field. As far as I understand it now, all coefficients and roots for all polynomials $f(x) \in F[x]$ lies in $F$. But what about the ...
5
votes
1answer
538 views

Galois group of $\mathbb{Q}[\sqrt{3},\sqrt{2}]$

I am trying to compute the Galois group of $\mathbb{Q}[\sqrt{3},\sqrt{2}]/\mathbb{Q}$ in the following way: First, $\mathbb{Q}[\sqrt{3},\sqrt{2}]/\mathbb{Q}$ is a Galois extension of the separable ...
0
votes
1answer
101 views

Tensor product of fields and homomorphisms

Question I'm pretty confused about changing the base field of a tensor product. For example, if $E/F,L/F$ are field extensions, the $E\otimes_F L$ could be seen as a vector space over $E$, by ...
3
votes
2answers
217 views

Subfield of $\mathbb{Q}[\zeta]$ fixed by conjugation

I am reading Milne's Fields and Galois Theory notes (I am reviewing these topics). On pages 39-40, the extension $\mathbb{Q}[\zeta]/\mathbb{Q}$ is analysed, where $\zeta$ is a primitive $7$th root of ...
3
votes
1answer
87 views

Show that $\mathbb{Q}(\zeta)$ is the Splitting Field for $x^n - 1 \in R_n[x]$

Let $R_n = \{\bar{x}$ modulo $n : (x,n) = 1\}$ which forms a group under multiplication. Let $p(x) = x^n - 1 \in \mathbb{Q}_n[x]$ have roots $\zeta_1, \zeta_2, \ldots , \zeta_n$. Prove that there is ...
4
votes
1answer
510 views

A finite field extension $K\supset\mathbb Q$ contains finitely many roots of unity

I must show that any finite field extension $K\supset \mathbb Q$ can only contain finitely many roots of unity. I reasoned in the following manner: Let $n<\infty$ be the degree of $K$ over ...
1
vote
1answer
51 views

What kind of field is denoted by $\mathbb{Z}_2^n$?

For a homework question I have to prove whether or not some set is a vector space over a field $\mathbb{Z}_2^n$, but I am not sure what this notation means and the textbook doesn't seem to clarify. ...
1
vote
0answers
157 views

Find the minimal polynomial

Let $K=\mathbb Q(\sqrt{-14})$. Then what is splitting field of $K(\sqrt{2+\sqrt{2}-1})$ over $\mathbb Q$ ?
0
votes
1answer
18 views

Cauchy sequence in valued fields

I can't understand this property, left unproved by my textbook as a trivial fact: let $K$ be a valued field, with valuation $\left|\phantom{x}\right|:K\longrightarrow\mathbb{R}$, let $\{a_n\}$ be a ...
8
votes
2answers
244 views

On irreducible factors of $x^{2^n}+x+1$ in $\mathbb Z_2[x]$

Prove that each irreducible factor of $f(x)=x^{2^n}+x+1$ in $\mathbb Z_2[x]$ has degree $k$, where $k\mid 2n$. Edit. I know I should somehow relate the question to an extension of $\mathbb Z_2$ of ...
6
votes
2answers
308 views

Spec of tensor product of fields

Suppose $K/k$ is a finite separable extension of degree $n$. How to show that there exists a finite separable extension $k'/k$ such that $\operatorname{Spec}(K \otimes_k k') $ consists of $n$ ...
1
vote
1answer
66 views

how these elements come? - finite field with polynomial $x^2+1$

I am given this finite field. $$K=\mathbb{Z_3}[x]/p$$ with $$p(x)=x^2+1$$ In my textbook, i see that the elements of this field are: $$\{0, 1, 2, x, x +1, x+2, 2x, 2x + 1, 2x + 2\}$$ I dont ...
1
vote
1answer
214 views

Finding the $6$ $\mathbb{Q}$ Automorphisms of $\mathbb{Q}[\sqrt[3]{2}, \omega]$

Let $F = \mathbb{Q}$ and consider $\mathbb{Q}[\sqrt[3]{2}, \omega] = E$, a $6$ degree extension of $F$ that splits $p(x) = x^3 - 2$. I am trying to establish that there are $6$ $F$ automorphisms of ...
0
votes
2answers
111 views

How can I find all elements of this field?

I am given this finite field. $$K=\mathbb{Z_3}[x]/p$$ where $p(x)=x^2 + 1$ How do I find all elements of this field?
2
votes
1answer
63 views

Determine if a field is a field of fractions

Let $R$ be a ring, let $K$ be a field, containing $R$ as a subring. Suppose that, for every $x\in K$, $x\neq 0$, either $x$ is in $R$ or $x^{-1}$ is in $R$. Can I conclude that $K$ is the field of ...
5
votes
1answer
448 views

The Galois group of a composite of Galois extensions

Morandi's Field and Galois Theory, exercise 5.19b Let $K$ and $L$ be Galois extensions of $F$. The restriction of function map, namely, $\sigma\mapsto(\sigma\vert_K,\sigma\vert_L)$ induces an ...
5
votes
2answers
65 views

How to think about the object $A\otimes_kk'$.

Let $k'/k$ be a finite extension of fields, and let $A$ be a finitely generated commutative $k'$-algebra. Through $k\hookrightarrow k'$ we can consider $A$ to be a finitely generated commutative ...
4
votes
1answer
50 views

Absolute value defined in a field

Let $\mathbb{K}$ be any field. Let $\left|\cdot\right|:\mathbb{K}\longrightarrow\mathbb{R}$ be a function which satisfies $\left|x\right|>0$ if $x\neq 0_{\mathbb{K}}$; $\left|0\right|=0$ ...
4
votes
1answer
177 views

The irreducibility of $x^p-a$ implies that of $x^{p^2}-a$

Morandi's Field and Galois Theorey, exercise 10.5c Let $p$ be a prime, and suppose that either $F$ contains a primitive $p$th root of unity for $p$ odd, or that $F$ contains a primitive fourth ...
3
votes
1answer
44 views

Is the category of ordered fields thin?

Pretty much everything I know about ordered fields, I learned in high school. So I apologize if this is a silly question. The category of fields is not thin. For example, complex conjugation is a ...
2
votes
1answer
138 views

Show that $ \sum q^{-\deg \ p(x)} $ diverges

Show that $\sum q^{-\deg \ p(x)}$ diverges, where the sum is over all monic irreducibles $p(x)$ in $K\left[x\right]$, where $k$ is finite field with $q$ elements. First show that $\sum q^{-\deg ...
1
vote
1answer
29 views

modular arithmetic over a field

Correct me if I am wrong here, but I just had a couple of questions that I wanted to make sure were right. First, consider $\mathbb{F}_{9} = \mathbb{F}_{3}[x] / (x^2 + 1)\mathbb{F}_{3}[x].$ Would ...
1
vote
3answers
307 views

Show that $f(x)=2x+1$ has a multiplicative inverse in $\mathbb{Z}_4[x]$, the integers mod 4.

Show that $f(x)=2x+1$ has a multiplicative inverse in $\mathbb{Z}_4[x]$, the integers mod 4. I don't really know where to start (besides dividing $1$ by $f(x)$). I thought having multiplicative ...
0
votes
1answer
120 views

List all polynomials of degree 3 in the field of integers mod 2 $\mathbb{Z}_2[x]$.

List all polynomials of degree 3 in the field of integers mod 2 $\mathbb{Z}_2[x]$. In $\mathbb{Z}_2[x]$, the only elements are 0,1, since everything above 1 equals either 0 or 1 in mod(2). I think ...
8
votes
1answer
218 views

When does $\|z^2\|=\|z\|^2$

Let $k \in \mathbb{Z}$ and consider the field extension $K := \mathbb{Q}[\sqrt{k}]$. Define a norm on $K$ given by $\|p+q\sqrt{k}\| := \sqrt{p^2+q^2}$. For any $z \in K$, I was interested to know when ...
4
votes
1answer
74 views

Proof of necessary condition for constructibility of a number

I'm reading a proof of the necessary condition for a real number to be constructible, and it seems to leave out a few details that I can't really fill in. This is what I understand so far. We have to ...