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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2
votes
1answer
30 views

How do we define discriminant over a commutative ring?

Let $f$ be a nonconstant polynomial over a field $F$. Since there exists a splitting field of $f$ over $F$, $f$ can be decomposed as $f=c\prod_{i=1}^n (X-\alpha_i)$ Hence, it is possible to define ...
2
votes
0answers
55 views

Galois group over $\mathbb{Q}$ [closed]

Let $$\begin{align*} K&=\mathbb{Q}(\{\text{all $2^n$-th roots of unity for $n\in\mathbb{N}$}\})\\ L&=\mathbb{Q}(\{\text{all $n$-th roots of unity for $n\in\mathbb{N}$}\}) \end{align*}$$ What ...
0
votes
0answers
27 views

Choice of Primitive Element in ''Primitive Element Theorem''

Let $F$ be a field of characteristic $0$, and $K$ a finite extension of $F$. Then it is well known (see this) that $K$ can be obtained by attaching an element $\alpha \in K$ to $F$, i.e. ...
2
votes
1answer
37 views

Relation between roots of an irreducible polynomial

Let $K$ be a field, and $\alpha$ be an element of a separable extension of $K$, such that $\alpha^p\in K$, but $\alpha\notin K$, $p$ a prime. Let $f$ be the minimal polynomial of $\alpha$ over $K$ ...
1
vote
1answer
25 views

Extension of field automorphism to automorphism of algebraic closure

Let $k$ be a field and let $f(x)\in k[x]$ be irreducible. Let $K$ be the algebraic closure of $k$, and say among the roots of $f(x)$ are $\alpha,\beta\in K$. Then there exists an automorphism of $K$ ...
2
votes
2answers
67 views

If $X^p-a$ has no zeros in a field $F$ of characteristic $p$ where $a \in F$, is it irreducible?

Let $F$ be a field of characteristic $p>0$ and $a\in F$. I have an easy question which I'm stuck on. If the polynomial $X^p-a$ has no zeros in $F$ then is it irreducible over $F$? ...
2
votes
1answer
47 views

Problem involving cubic field extensions

Let $F$ be a field of characteristic $0$ and let $L$ be a cubic extension. I want to show that there exists an element $a \in F,$ and an extension $L_0$ of $\mathbb{Q}(a)$ such that ...
3
votes
4answers
51 views

Establishing additive and multiplicative inverses for a finite field

I am struggling with the following problem: Let $F$ be a finite field, and let $G$ be a subset of $F$ with the following properties: $0$ and $1$ are in $G$; whenever $a$ and $b$ are in $G$, $a + ...
2
votes
1answer
35 views

Why is the degree of this field extension $[K(x,y): K(x^p, y^p)]= p^2$?

Fix a prime $p$. Let $K:=\overline{\mathbb{F}}_p$ be the algebraic closure of $\mathbb{F}_p$. Consider now the field $K(x,y)$ of rational functions in $x,y$ and its subfield $K(x^p,y^p)$ of rational ...
2
votes
2answers
48 views

If $A\otimes_k l$ is a normal integral domain then $K(A)\otimes_k l$ is a field.

I am trying to solve Ex. 5.4.M in Vakil's notes. Quoting the text: Suppose $A$ is a $k$-algebra, and $l/k$ is a finite extension of fields. (Most likely your proof will not use finiteness; this ...
1
vote
6answers
78 views

Constructing $\mathbb{C}$ from $\mathbb{R}$

I'm having difficulty grasping the notion that you can define the complex numbers as $\mathbb{C}=\mathbb{R}[t]/\langle t^2+1\rangle$. As far as I understand, $\mathbb{R}[t]$ is the set of all ...
3
votes
0answers
42 views

Ordered fields are real?

This question is motivated by another question posted earlier today. Let $F$ be a field endowed with an embedding $F\hookrightarrow\Bbb R$. Then the ordering of $\Bbb R$ induces an ordering on $F$. ...
0
votes
2answers
28 views

Field $F$ with $\operatorname{char}F=3$ and algebraic over $\mathbb{F}_3$ has a primitive root of unity.

Suppose that $F$ is a field with $\operatorname{char}F=3$ and $F$ is an algebraic extension of $\mathbb{F}_3$. Prove that $F$ contains a primitive $n$th root of unity for some $n>2$.
1
vote
1answer
40 views

A polynomial's irreducibility in $\Bbb{Z}_p$

Show that if $f$ is irreducible in $\Bbb{Z}_p[x]$ then $f$ divides $x^{p^n} - x$ for some $n \in N$. I know that: $f$ is irreducible, so $F = \Bbb{Z}_p / {\left\langle f\right\rangle}$ is a ...
0
votes
1answer
41 views

The characteristic of real-closed fields is zero?

We know that $F$ is a real-closed field if $F$ is not algebraically closed but $F(\sqrt{-1})$ is algebraically closed. So I have this question What can we say about $\operatorname{char}F$? Is it ...
2
votes
2answers
42 views

$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
0
votes
1answer
35 views

If $F$ is a field, then any two algebraic closures are isomorphic by an isomorphism that is the identity on $F$.

To start, suppose $K_1$ and $K_2$ are two algebraic closures of $F$. (a) Let $P$ be the set of partial functions $f$ from $K_1$ to $K_2$ with the following properties: $F$ is contained in ...
0
votes
1answer
26 views

Do real quadratic fields with unique primary factorization exist?

Bumped in Stillwell's book "Elements of Number Theory" into "The real quadratic fields with unique prime factorization are still not known ...". But doesn't $\mathbb{Q}[\sqrt{2}]$'s ring of integers ...
3
votes
1answer
71 views

Finding the Fixed Fields in the Galois Correspondence for the Splitting Field of $x^4-3x^2+4$ over $\mathbb{Q}$

I have found the Galois group of the polynomial $x^4 - 3x^2 + 4$ (see below), but I am not sure how to find the fixed fields in the Galois correspondence. The roots of the polynomial are $$\pm ...
0
votes
0answers
24 views

Question on primitive element for extensions of finite fields [duplicate]

I was given this question in field theory class which I am stumped on as after some effort I still cannot tackle. I am given a general natural number $n>1$ and am asked to prove or disprove: ...
2
votes
1answer
35 views

Another abstract algebra/field theory question

Suppose that $F$ is a field, $S \subseteq F^n$ and $I$ is an ideal in $F[x_1, \cdots, x_n] = F[\bar{x}]$. Define $$I(S) = \{ f \in F[\bar{x}]: f(\bar{s}) = 0, \forall \bar{s} \in S\}$$ and ...
4
votes
1answer
40 views

splitting field of $x^8-1$ over $\mathbb F_3$

Suppose $F=\mathbb F_3$ and $f(x)=x^8-1$ in $F[x]$. I tried finding the Galois group of the splitting field of $f(x)$ over $F$ and I'm not so sure if what I did was correct. I began by looking at ...
1
vote
0answers
27 views

Subgroups of the multiplicative group of a Field [duplicate]

I am reading S. Roman's book "Field Theory". In chapter $1$ I found the following exercise Let $F^*$ be the multiplicative group of all nonzero elements of a field $F$. We have seen that if G is a ...
4
votes
2answers
54 views

primitive element $a$ of $\mathbb F_{p^n}/\mathbb F_p$ such that $a^n\in\mathbb F_p$

Is it true that for every $n\in \mathbb N$ there exists a prime $p$ such that the extension $\mathbb F_{p^n}/\mathbb F_p$ has a primitive element $a\in \mathbb F_{p^n}$ and $a^n\in\mathbb F_p$? I ...
4
votes
2answers
52 views

Showing that the Field Extension $\mathbb{Q}(T^{1/4})/ \mathbb{Q}(T)$ is not Galois

Prove that $\mathbb{Q}(T^{1/4})$ is not Galois over $\mathbb{Q}(T)$, where $T$ is an indeterminate. I am not sure how to proceed due to the indeterminate. It suffices to show that the degree of ...
2
votes
0answers
53 views

A field F can be extended to one in which all polynomials over F have a solution

Please check to see if my methods are correct. This question is a lot to absorb all at once. Let P be the set of non-constant polynomials over a field F and let X be the set of finite subsets of ...
0
votes
1answer
39 views

Field extension generated by $\alpha$ and separability

In my notes, I have written that the field extension of $k$ generated by an element $\alpha \in K$, where $K$ a larger field, is defined to be $$k(\alpha) = \bigcap_{ \alpha \in E} E$$ where $E ...
5
votes
2answers
37 views

Galois group and degree of splitting field over complex rational functions.

Suppose $F=\mathbb C (t)$ the field of rational functions over $\mathbb C$. let \begin{equation*}f(x)=x^6-t^2\in F[x]\end{equation*}Denote $K$ as the spliting field of $f$ over $F$. I'm trying to ...
2
votes
2answers
50 views

The Galois group of a specific polynomial $ f(x) = x^6-2x^3+2 \in Q[x] $

Hello all I was given this question in Algebra asking me to show a polynomial's Galois group has both a subgroup and quotient group isomorphic to $S_3$ the three symmetric group. The polynomial in ...
0
votes
2answers
42 views

Example check: two algebraically closed fields with one a subset of the other

The problem asks to find two different algebraically closed fields $\mathcal{E}$ and $\mathcal{F}$ with $\mathcal{E} \subseteq \mathcal{F}$. We have not done a whole lot of stuff with algebraically ...
1
vote
2answers
39 views

What is the characteristic of a field in lay terms?

And why do we usually assume a field has characteristic not equal to 2?
3
votes
0answers
15 views

Uses for coefficients other than the trace and norm for an element belonging to a field.

The trace and norm are very useful as maps from a field extension to the base field since they are multiplicative/additive and have a lot of other nice properties. They can be defined as two of the ...
3
votes
1answer
35 views

Question on finite fields and their extensions

I have been given this question in Algebra class on finite fields which I have tried to solve but to no avail, so all help appreciated. I am given $ p=13;q=p^6 $, then I am asked to prove or give a ...
3
votes
0answers
22 views

$O_S$ is the integral closure of $k[T]$ in $F$ for some embedding of $k(T)$ in $F$?

Let $F$ be a function field in one variable over a field $k$. Let $S$ be a nonempty finite subset of the set of all places of $F$. Let$$O_S = \{f \in F: \text{ord}_v(f) \ge 0 \text{ for all }X ...
2
votes
0answers
31 views

Calculating the Galois group of a covering map

Suppose $C$ is an algebraic curve and $\phi:C\rightarrow \mathbb{P}^{1}$ is a covering map of the complex projective line ramified at $\{0,1,\infty\}$ only. Suppose $\phi':C'\rightarrow ...
3
votes
1answer
45 views

Given $K(\alpha)/K$ and $K(\beta)/K$ abelian extensions, prove that $K(\alpha + \beta)/K$ is an abelian extension.

Problem: Let $K(\alpha)/K$ and $K(\beta)/K$ algebraic field extensions so that their respective Galois groups are abelian. Prove that the Galois group of the field extension $K(\alpha + ...
1
vote
2answers
42 views

Let $K=GF(2)$ and $p(x)= x^3 + x+1.$ Show that $p$ is irreducible in $K[x]$

Let $K=GF(2)$ and $p(x)= x^3 + x+1$ Show that $p$ is irreducible in $K[x]$ First of all am I right in interpreting: $$GF(2) = \mathbb Z / 2 \mathbb Z= \{ 0,1\}$$ So basically, $p(x)$ is a ...
2
votes
1answer
61 views

All primes $p$ for which $x^{2} +1$ irreducible over $\mathbb{F}_{p}$

Let $\mathbb{F}_{p}$ be the field with $p$ elements. Determine all primes $p$ for which $x^{2}+1$ is irreducible over $\mathbb{F}_{p}$. Here's what I've got. I think this might be finished but I'm ...
1
vote
6answers
147 views

In a field $F=\{0,1,x\}$, how does $1 + 1 = x$?

I understand that in a field with two elements $1 + 1 = 0$, but in a field with three I do not understand how $1 + 1 =x$. The work I have done so far is: \begin{align} 1 + 1 &= \{ 0 , 1 , x\}\\ 1 ...
11
votes
1answer
229 views

Galois Group of $x^{4}+7$

I am trying to find the Galois group of $x^{4}+7$ over $\mathbb{Q}$ and all of the intermediate extensions between $\mathbb{Q}$ and the splitting field of this polynomial. I have found that the ...
6
votes
2answers
73 views

Embedding $\mathbb{F}_{q^2}^*$ into $GL_2(\mathbb{F}_{q})$

If we see $\mathbb{F}_{q^2}$ as a $2$-dimensional vector space over $\mathbb{F}_{q}$ (and pick a base) then we can identify $\operatorname{Aut}_{\mathbb{F}_{q}}(\mathbb{F}_{q^2})$ with ...
8
votes
2answers
72 views

Irreducibility of cyclotomic polynomials over number fields

Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. For a positive integer $n$, let $\Phi_n(X)$ denote the $n$-th cyclotomic polynomial. Is it possible to say that there exist at ...
0
votes
1answer
26 views

Radical extension and algebraic solution of an irreducible polynomial

Suppose that $k$ is a field with characteristic equal to zero, that $P \in k[X]$ is an irreducible polynomial and that $\alpha$ is a root of $P$ in an algebraic closure $\overline{k}$. Suppose also ...
1
vote
1answer
39 views

Every irreducible polynomial f over perfect field F is separable

Every irreducible polynomial f over perfect field F is separable. Can you check my proof? Let f is inseparable. So we have $f=\sum_i h_ix^i$ and $f^p=\sum_i h_i^px^{ip}$ Now I use Frobenius mapping ...
1
vote
1answer
39 views

Is every field a Krull domain?

Background: A Krull domain is an integral domain $A$ with a family $(v_i)$ of valuations on the field of fractions $K$ for $A$ satisfying the following conditions: Each $v_i$ is discrete. The ...
1
vote
1answer
55 views

If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?

Let's consider a field $F$ and $\alpha,\beta\in\overline{F}-F$ (where $\overline{F}$ is an algebraic closure). If $F(\alpha)=F(\beta)$, is it true that $\alpha$, $\beta$ have the same minimal ...
2
votes
0answers
40 views

Calculus on Spaces other than $R$

I was thinking that all that the real numbers (and the complex numbers similarly) need for calculus to work is for them to be a field and a complete metric space, and probably have the usual field ...
0
votes
1answer
57 views

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$ My answer: Must show: i) $0 \in \mathbb{Q}{[\sqrt3]}$ ii) $1 \in ...
5
votes
1answer
58 views

Does it make sense to talk about complex matrices over the field of real numbers, R?

I don't see an issue with considering a vector space of complex matrices over R -- addition of matrices makes sense, but scalar multiplication will be done with real numbers. But I wanted to ask, ...
2
votes
1answer
43 views

extend trace and norm from a number field $K$ to $ K \otimes_{\mathbb Q} \mathbb R $

I read somewhere that we can extend the trace and the norm of a number field $K$ to the commutative algebra $ V=K \otimes_{\mathbb Q} \mathbb R$. Before state exactly my question, let me write the ...