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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3
votes
1answer
133 views

Is an automorphism of a normal extension determined by its image of the maximal separable sub extension?

Let $L / K$ be a normal, algebraic field extension. Suppose that the maximal separable sub- extension $M/K$ is finite, $K \subseteq M \subseteq L$. By the primitive element theorem, $M=K(x)$ for some ...
0
votes
2answers
112 views

Question about isomorphism between a ideal and a polynomial ring

Sorry for my ignorance, my question is: Let be $F[X]$ a polynomial quotient ring, where $F$ is a finite field with characteristic 2. Are there any ideal, $I$, such that $I$ is isomorphic to $F[X]$?.
1
vote
1answer
92 views

Question about a corollary about Finite Fields

Definition: A field extension $E$ of $F$ is of degree $n$ (and is called a finite field extension) if $E$ is an $n$-dimensional vector space over $F$. Theorem: Let $E$ be a degree $n$ finite ...
4
votes
1answer
130 views

Unramified extension is normal if it has normal residue class extension

Let $K/F$ be an unramified extension such that $\rho_K / \rho_F$ (the corresponding extension of residue classes) is normal. Prove $K/F$ is normal. I guess I need to do some polynomial lifting, but ...
0
votes
1answer
150 views

What is a valuation associated to an ordering on a field?

If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined? I was searching through Prestel & Delzell's Positive ...
4
votes
1answer
332 views

A formula for the roots of a solvable polynomial

Let $F$ be a field and $p(x)\in F[x]$ a separable polynomial, denote $K$ as the splitting field of $p$ and assume that $K/F$ is Galois with a solvable Galois group. I don't understand if this imply ...
2
votes
2answers
155 views

Is a polynomial solvable by roots iff every irreducible factor is?

Let $F$ be a field, I asked myself if $p(x)\in F[x]$ is solvable by radicals iff every irreducible factor is solvable by radicals. My thoughts: If every irreducible factor is solvable by roots then ...
3
votes
1answer
286 views

Irreducible Polynomials in Finite Fields

I'm reading through some notes online concerning finite fields, and attempting to come up with a proof that all finite fields of the same size are isomorphic. But I'm getting stuck at a certain point, ...
0
votes
1answer
53 views

Basic question about fractions

I'm solving some exercises about fields and am trying to find the inverse for $a_1 + \sqrt{2}b_1$, i.e. $\frac{1}{a_1 + \sqrt{2}b_1}$. This means I need to split the fraction into something of the ...
4
votes
4answers
181 views

How many elements in the finite field $F_{256}$ satisfy $x^{103}=x$?

How many elements of the finite field $\mathbb{F}_{256}$ with 256 elements satisfy $x^{103}=x$?
3
votes
1answer
310 views

The order of the Galois group of a cyclotomic field over a finite prime field [duplicate]

Possible Duplicate: For what $(n,k)$ there exists a polynomial $p(x) \in F_2\[x\]$ s.t. $\deg(p)=k$ and $p$ divides $x^n-1$? Galoisgroup $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} ...
0
votes
1answer
858 views

Cyclotomic polynomial over a finite prime field [duplicate]

Possible Duplicate: Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})\[X\]$ Let $p$ be a prime number. Let $l$ be an odd prime number such that $l \neq p$. Let $X^l - 1 \in ...
3
votes
1answer
568 views

Subgroup of Galois group of polynomial over $\mathbb{Q}$

Let $K$ be the splitting field of $x^5-3 \in \mathbb{Q}[x]$. We can see $K = \mathbb{Q}(3^{1/5}, \zeta_5)$ where $\zeta_5 = e^{2 \pi i/5}$, and $[K: \mathbb{Q}] = 20$. It's easy to see $\sqrt{5} \in ...
3
votes
2answers
340 views

Proving there are no subfields

I am trying to solve Q11 at pg. 582 from the book Abstract algebra by Dummit and Foote, the question is: Let $f\in\mathbb{Z}[x]$ be an irreducible quartic whose splitting field has Galois group ...
6
votes
1answer
465 views

Positivity of the norm of an element of a cyclotomic field

Let $l$ be an odd prime number and $\zeta$ be an $l$-th primitive root of unity in $\mathbb{C}$. Let $\mathbb{Q}(\zeta)$ be the cyclotomic field and $\alpha$ be a non-zero element of ...
1
vote
1answer
97 views

If $K,E$ are subfields of $\Omega/F$ then $KE/F$ is a finite Galois imply $K/K\cap E$ is Galois?

Let $\Omega/F$ be a field extension and $K,E$ be two subfields of $\Omega/F$. Assume that $KE/F$ is a finite Galois. I have a theorem in my lecture notes that claim $\text{Gal}(KE/E)\cong ...
1
vote
4answers
84 views

If $K/F$ is Galois and $E$ is a subextension then $E$ is generated by roots of a polynomial over $F$?

Let $K/F$ be finite Galois field extension, then $K$ is the splitting field of a separable polynomial $p$ over $F$, i.e. $K=F(a_{1},..a_{n})$ where $p=(x-a_{1})...(x-a_{n})$. My question is: is it ...
1
vote
2answers
5k views

Galois Field GF(4)

Question: Why is the table of GF(4) look like the one below? I know it has to do with the fact that 4 is composite Let GF(4) = {0,1,B,D} Addition: $$ \begin{array}{c|cccc} + & 0& 1& ...
4
votes
1answer
221 views

Finding quadratic extensions in $\mathbb{Q}\left(\sqrt{i+2}\right)$

Let $\alpha=\sqrt{i+2}$ and let $F=\mathbb{Q}\left(\alpha\right)$. Note that $\left[F:\mathbb{Q}\right]=4$ since the minimal polynomial of $\alpha$ over $\mathbb{Q}$ is $x^{4}-4x^{2}+5$. Show ...
4
votes
2answers
436 views

A question about a proof of a weak form of Hilbert's Nullstellensatz

I'm trying to prove the following (corollary 5.24 page 67 in Atiyah-Macdonald): Let $k$ be a field and let $B$ be a field that is a finitely generated $k$-algebra, i.e. there is a ring homomorphism ...
3
votes
2answers
494 views

A question about a weak form of Hilbert's Nullstellensatz

Corollary 5.24 on page 67 in Atiyah-Macdonald reads as follows: Let $k$ be a field and $B$ a finitely generated $k$-algebra. If $B$ is a field then it is a finite algebraic extension of $k$. We know ...
0
votes
1answer
64 views

Proving that $\Phi_{n}$ is irreducible (a problem with the proof)

I am trying to follow the proof in the book Abstract Algebra by Dummit and Foote (Theorem 41, pg. 554) that $\Phi_n$ is an irreducible monic polynomial in $\mathbb{Z}[x]$ of degree $\varphi(n)$. What ...
1
vote
1answer
59 views

Why $g(x^{p})=(g(x))^{p}$ in the reduction mod $p$?

In one of the proof in the book "Abstract Algebra'' by Dummit and Foote (Theorem 41, pg. 554) we have a monic polynomial $g(x)\in\mathbb{Z}[x]$, and the book claims that $g(x^{p})=(g(x))^{p}\mod p$ ...
0
votes
3answers
218 views

$x$ algebraic over $K$, $v$ a polynomial in $x$ then $v$ algebraic?

In the proof of proposition 5.23 Atiyah-Macdonald on page 66 use that if $x$ is algebraic over $K$ and $v = a_n x^n + \dots + a_1 x + a_0$ then $v$ is algebraic over $K$ (where $K$ is the field of ...
4
votes
2answers
146 views

Proving that if $\mathrm{char}(F)=p>0$ then if $g(x)\in F[x]$ is irreducible then $g(x)$ have multiple roots iff $g'(x)=0$

I am going over my lecture notes in my Field theory class and I saw this following statement without a proof: if $\mathrm{char}(F)=p>0$ then if $g(x)\in F[x]$ is irreducible then $g(x)$ have ...
4
votes
1answer
217 views

Showing existence of a field extension of degree $n$ for a finite field $F$

EDIT: Just mentioning that this is a homework question. This is my first time posting a question on math.stackexchange, so I hope you find it in your hearts to forgive any stylistic or rule ...
3
votes
1answer
171 views

Additive inverse

Let $F$ be the set of $\alpha\subset \mathbb{Q}$ with following properties. (I) $\alpha ≠ \emptyset$ and $\alpha ≠ \mathbb{Q}$ (II) $p\in \alpha$ and $q<p$ ⇒ $q\in \alpha$ (Notice that it's ...
2
votes
1answer
128 views

If the polynomial $f$ is zero on the nonzero set of another polynomial $g$, does $f=0$?

Suppose $f(x_1,\dots,x_n)$ is a polynomial in $n$ indeterminates over an infinite field $F$. Suppose $f((a_i))=0$ for all $n$-tuples $(a_i)$ such that $g((a_i))\neq 0$, where $g(x_1,\dots,x_n)$ is ...
23
votes
2answers
943 views

Why algebraic closures?

Let me begin by summarizing the question: Why do we care about fields closed under rational exponentiation, and less about fields closed under other operations? Historically the solution for ...
0
votes
2answers
130 views

Field of algebraic reals over the rationals

Let $L$ be the subfield of $\mathbb{R}$, of all reals that are algebraic over $\mathbb{Q}$: $L = \{ x\in \mathbb{R} : x \text{ is algebraic over } \mathbb{Q} \}, \;\;\; \mathbb{Q} \subseteq L$. Let ...
8
votes
1answer
398 views

Problem 18.7 in I. Martin Isaacs' Algebra

I am trying to solve the following problem in I. Martin Isaacs' Algebra: A graduate course, p.290: Let $f(X),g(X) \in F[X]$ and suppose $E \supseteq F$ is the splitting field both for $f(X)$ and ...
2
votes
1answer
91 views

Is there a necessary and sufficient condition to determine the generators of $\mathbb{Z}_p^\times$?

This is something I was wondering about. I know that the generators of the cyclic groups $(\mathbb{Z}_n,+)$ are precisely those integers coprime to $n$, and there are $\phi(n)$ of them. Now the ...
8
votes
4answers
882 views

Why is the product of all units of a finite field equal to $-1$?

Suppose $F=\{0,a_1,\dots,a_{q-1}\}$ is a finite field with $q=p^n$ elements. I'm curious, why is the product of all elements of $F^\ast$ equal to $-1$? I know that $F^\ast$ is cyclic, say generated by ...
3
votes
1answer
170 views

Problem 18.1 in I. Martin Isaacs' Algebra

I am trying to prove the following: Let $E/F$ be an arbitrary extensions. Show that $E/F$ is normal if and only if $E$ is the union of all those intermediate fields $K$ such that $K$ is the splitting ...
3
votes
1answer
334 views

Two notions of uniformizer

Let $X$ be a projective algebraic curve and consider a `uniformizing' map $h:X \rightarrow \mathbb{P}^1$. Is there any connection between this notion of uniformizer and a uniformizer of the maximal ...
10
votes
3answers
271 views

When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes?

I've been considering the rings $R_1=\mathbb{F}_p[x]/(x^2-2)$ and $R_2=\mathbb{F}_p[x]/(x^2-3)$, where $\mathbb{F}_p=\mathbb{Z}/(p)$. I'm trying to figure out if they're isomorphic (as rings I ...
2
votes
3answers
112 views

Is there a good way to solve for the inverse of $(u^2-u+4)$?

I'm having trouble calculating the inverse of a polynomial. Consider the polynomial $f(x)=x^3+3x-2$, which is irreducible over $\mathbb{Q}$, as it has no rational roots. So $\mathbb{Q}[x]/(f(x))\simeq ...
3
votes
1answer
313 views

If $F$ is a formally real field then is $F(\alpha)$ formally real?

Let us call a field $F$ $\textit{ formally real }$ if $-1$ is not expressible as a sum of squares in $F$. Now suppose $F$ is a formally real field and $f(x)\in F[x]$ be an irreducible polynomial of ...
4
votes
1answer
789 views

Finding a primitive element for the field extension $\mathbb{Q}(\sqrt{p_{1}},\sqrt{p_{2}},\ldots,\sqrt{p_{n}})/\mathbb{Q}$

Let $p_1,\ldots,p_n\in\mathbb{N}$ be different prime numbers, it can be shown that $[\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n}):\mathbb{Q}]=2^n$ and in any case it is clearly finite since ...
1
vote
3answers
878 views

How to think of the field $F(\alpha)$

The way I learned it was given a field extension $F \subset E$, and an element $\alpha \in E$ $$F(\alpha) := \{p(\alpha)/q(\alpha) : p(x), q(x) \in F[x] ,q(\alpha) \not = 0\} $$ Is there an easier ...
2
votes
1answer
86 views

Splitting field and subextension

Definition: Let $K/F$ be a field extension and let $p(x)\in F[x]$, we say that $K$ is splitting field of $p$ over $F$ if $p$ splits in $K$ and $K$ is generated by $p$'s roots; i.e. if ...
1
vote
2answers
401 views

If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$

If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$. A finite field is a simple extension of each of its subfields and $\mathbb{Z}_p$ is a subfield of every finite ...
2
votes
1answer
133 views

Normal Field Extension

$X^4 -4$ has a root in $\Bbb Q(2^{1/2})$ but does not split in $\Bbb Q(2^{1/2})$ implying that $\Bbb Q(2^{1/2})$ is not a normal extension of $\Bbb Q$ according to most definitions. But $\Bbb ...
2
votes
2answers
116 views

$f$ is irreducible in $\Bbb F[x]$

Let $\Bbb F$ be a field of characteristic $p\gt 0$ and $f(x)=x^{p^n}-c \in\Bbb F[x]$ where $n$ is a positive integer. If $c \notin \{a^p:a\in \Bbb F \}$, show that $f$ is irreducible in $\Bbb ...
2
votes
1answer
198 views

A property equivalent to $K$ being a finite-dimensional, normal extension field of $F$

Prove that a finite-dimensional extension field $K$ of $F$ is normal if and only if it has this property: Whenever $L$ is an extension field of $K$ and $\sigma :K\rightarrow L$ an injective ...
2
votes
1answer
152 views

Number of elements in $\mathbb{Z}_p[x]/ \langle f \rangle$

I want to determine the number of elements in $\mathbb{Z}_p[x]/ \langle f \rangle$ where $f \in \mathbb{Z}_p[x]$ is an irreducible polynomial with $k$ degree bigger than 2. Is the number of elements ...
29
votes
1answer
2k views

Is there a purely algebraic proof of the Fundamental Theorem of Algebra?

Among the many techniques available at our disposal to prove FTA, is there any purely algebraic proof of the theorem? That seems reasonably unexpected, because somehow or the other we are depending ...
3
votes
3answers
893 views

Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$

Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$. Ok so originally I messed around with $x^3 + x +1$ for a bit looking for an easy way to factor it and eventually decided that the ...
1
vote
2answers
2k views

Is vector space a field? Or more than that?

As you all know, vector space is closed under scalar multiplication, scalar product, vector product and addition. If I take scalar product, vector space is a field, but if i take vector product, ...
6
votes
1answer
613 views

Calculating the norm of an element in a field extension.

Given a number field $\mathbb{Q}[\beta]$, where the minimal polynomial of $\beta$ in $\mathbb[Z][x]$ has degree $n$, I would like to calculate the norm of the general element ...