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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field of algebraic numbers

Would anyone be able to give me an outline or a hint towards the proof that the field of algebraic numbers is an infinite extension of the field of rationals? Many Thanks
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2answers
157 views

Closure of a number field with respect to roots of a cubic

Consider the following property of subfields ${\mathbb K}$ of ${\mathbb C}$ : $$ \text{Any polynomial of degree 3 with coefficients in} \ {\mathbb K} \ \text{has a root in } {\mathbb K} \ \ \ \ ...
3
votes
1answer
969 views

Sums and products of algebraic numbers

How does one go about proving that the sums and products of two algebraic numbers over a field $F$ (say $a,b\in K$, where $K/F$ is a field extension) is also algebraic? If we call $f_a$ and $f_b$ ...
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4answers
4k views

Finding inverse of polynomial in a field

I'm having trouble with the procedure to find an inverse of a polynomial in a field. For example, take: In $\frac{\mathbb{Z}_3[x]}{m(x)}$, where $m(x) = x^3 + 2x +1$, find the inverse of $x^2 + ...
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3answers
232 views

is it possible for two fields with the same characteristic to not be isomorphic?

I know there is a long list of invariants, but is there a proof or theorem that explains two fields with the same characteristic are isomorphic?
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4answers
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Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
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4answers
4k views

Finding the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$.

I have to find the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$. The suggested way of doing it is to prove that $\mathbb Q[\sqrt 2 + \sqrt[3] 2]=\mathbb Q[\sqrt 2,\sqrt[3] 2]$ first. ...
26
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3answers
2k views

Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”?

The function $f(x)=x+\sin(x)$ is easily checked to be a bijection from the reals to itself, and so it has a unique inverse $y\mapsto g(y)$ such that $f\circ g=g\circ f$ are both the identity map. ...
11
votes
1answer
274 views

Field reductions

If there is a field $F$ that is a field reduction of the real numbers, that is $F(a)=\mathbb{R}$ for some $a$, let's also denote this $F=\mathbb{R}(\setminus a)$, then given $x \in \mathbb{R}$ is ...
16
votes
5answers
333 views

Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we ...
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votes
5answers
1k views

Infinite Degree Algebraic Field Extensions

In I. Martin Isaacs Algebra: A Graduate Course, Isaacs uses the field of algebraic numbers $$\mathbb{A}=\{\alpha \in \mathbb{C} \; | \; \alpha \; \text{algebraic over} \; \mathbb{Q}\}$$ as an example ...
6
votes
1answer
375 views

Degree of field extension

If $p$ is odd prime and $c=\cos(\frac{2\pi}{p})$, $s=\sin(\frac{2\pi}{p})$ then for which values of $p$ does $\mathbb{Q}(s,c)=\mathbb{Q}(c)$?
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votes
2answers
437 views

isomorphism of Dedekind complete ordered fields

In the last chapter of Spivak's Calculus, there is a proof that complete ordered fields are unique up to isomorphism. I find the first steps in it somewhat suspicious. Specifically, I believe he is ...
23
votes
3answers
482 views

On the meaning of being algebraically closed

The definition of algebraic number is that $\alpha$ is an algebraic number if there is a nonzero polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $p(\alpha)=0$. By algebraic closure, every nonconstant ...
10
votes
3answers
194 views

Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?

We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
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votes
2answers
380 views

Why aren't there any coproducts in the category of $\bf{Fields}$?

Well the question is stated in the title. I dont know much about field theory and i was suprised when i read it on wikipedia please provide some examples thanks in advance
4
votes
2answers
283 views

What is the condition for a field to make the degree of its algebraic closure over it infinite?

As we all know, the algebraic closure often has an infinite degree. Also, this shows the necessary and sufficient condition for a Galois extension to be a finite extension of fields. However, we may ...
3
votes
1answer
127 views

When is a divisible group a power of the multiplicative group of an algebraically closed field?

It is known that for any algebraically closed field $\mathbb{F}$ its multiplicative group $\mathbb{F}^*$ is a divisible group, and consequently any power $\mathbb{F}^*\times\cdots\times \mathbb{F}^*.$ ...
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votes
2answers
391 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 ...
2
votes
2answers
596 views

Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$

$d \in \mathbb{Z}$ is a square-free integer ($d \ne 1$, and $d$ has no factors of the form $c^2$ except $c = \pm 1$), and let $R=\mathbb{Z}[\sqrt{d}]= \{ a+b\sqrt{d} \mid a,b \in \mathbb{Z} \}$. ...
2
votes
1answer
291 views

a property of the radical closure of a field

Definition 1: Let $K$ be a field. If $\alpha$ is an algebraic element over K, such that $\alpha^n \in K $ and such that $x^n-\alpha^n \in K[x] $is also irreducible over K. Then we call $ K(\alpha)$ a ...
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vote
1answer
457 views

If a normal $K/F$ has no intermediate extensions, then $[K : F]$ is prime

Let $K$ be a finite normal extension of $F$ such that there are no proper intermediate extensions of $K/F$. Show that $[K:F]$ is prime. Give a conterexample if $K$ is not normal over $F$.
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3answers
259 views

Proving that $-a=(-1)\cdot a$

As the title reveals, I want to prove (based on the axioms of field) that $$-a=(-1)\cdot a$$ I've been trying for a while now, but couldn't think of a way to do it and it got me thinking that maybe ...
7
votes
1answer
205 views

Field Extensions

Let $L/K$ a finite extension and $f(x)\in K[x]$ a non-linear irreducible polynomial. Prove that if $\mathrm{gcd}\left( \mathrm{deg}(f) , \left[ L:K \right] \right)=1$ then $f(x)$ has no roots in ...
6
votes
1answer
278 views

Generating Elements of Galois Group

I am trying to prove the following: Let $K=\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n})$. Show that $K/\mathbb{Q}$ is Galois with Galois group $(\mathbb{Z}/2\mathbb{Z})^n$. I have attached my proof ...
4
votes
1answer
244 views

A proof of the Lagrange's theorem on cyclic extention fields

I came up with the following proof of the above theorem which is the key to the Galois's theory of algebraic equations. The usual proof uses Lagrange resolvent or Hilbert 90 which uses a similar ...
4
votes
2answers
686 views

Exercise on separable polynomials over fields of prime characteristic

Having learned about separable polynomials today in class, I tried to do the following exercise concerning separable polynomials, namely: Suppose $f$ is the minimal polynomial of $a$ over a field ...
4
votes
3answers
1k views

Example of a complete, non-archimedean ordered field

I'm looking for a concrete example of a complete (in the sense that all Cauchy sequences converge) but non-archimedean ordered field, to see that these two properties are independent (an example of ...
3
votes
2answers
98 views

Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?

As title, Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated? Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, ...
3
votes
1answer
933 views

What's the difference between hyperreal and surreal numbers?

The Wikipedia article on surreal numbers states that hyperreal numbers are a subfield of the surreals. If I understand correctly, both fields contain: real numbers a hierarchy of infinitesimal ...
3
votes
1answer
217 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 ...
3
votes
3answers
685 views

Primitive polynomials of finite fields

there are two primitive polynomials which I can use to construct $GF(2^3)=GF(8)$: $p_1(x) = x^3+x+1$ $p_2(x) = x^3+x^2+1$ $GF(8)$ created with $p_1(x)$: 0 1 $\alpha$ $\alpha^2$ $\alpha^3 = ...
2
votes
2answers
317 views

Compositum of abelian Galois extensions is also?

Suppose I have a field $k$ and two extensions $K/k$ and $L/k$ which are both abelian Galois extensions of $k$. Then (assuming $K$ and $L$ are both contained in some bigger field) is the compositum ...
1
vote
1answer
52 views

Reducibility over a certain field.

Let $K=F_2[x]/(x^3+x+1)$. I want to show that $f(x)=x^4+x^2+1$ is reducible over $K$ but has no roots in it. How to proceed? I know that $F$ contains 8 elements, how is the structure of these ...
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votes
3answers
470 views

Showing a homomorphism of a field algebraic over $\mathbb{Q}$ to itself is an isomorphism.

Suppose $F$ is algebraic over $\mathbb{Q}$ and $\varphi : F\to F$ is a homomorphism. Prove $\varphi$ is an isomorphism. Showing injectivity follows from the fact that the only ideals in a field ...
6
votes
2answers
297 views

Field Isomorphisms

Suppose $F/L$, $F'/L$, $L/K$ finite extensions of fields. If $F$, $F'$ isomorphic over $K$ then does it follow that they are isomorphic over $L$? I think probably not, but I can't come up with a ...
6
votes
3answers
228 views

What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$?

Sorry to ask this, I know it's not really a maths question but a definition question, but Googling didn't help. When asked to show that elements in each are irreducible, is it the same?
6
votes
1answer
341 views

Does $\varphi(1)=1$ if $\varphi$ is a field homomorphism?

Is it by definition that $\varphi(1)=1$ if $\varphi$ is a field homomorphism ? My field theory lecture said that yes, but now $\varphi\equiv0$ is not a field homomorphism... What is the 'standard' ...
5
votes
1answer
300 views

Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$

Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$. My justification for this question is as follows; Suppose $F(\alpha^2)\subsetneq F(\alpha)$, we have $F \subsetneq F(\alpha^2) ...
5
votes
2answers
168 views

Towers of perfect fields of mostly order $p^t$

This question is strongly related to this question. I'm curious about the following result. Fix a prime $p$ and let $L$ be a perfect field and $q_1(t),\dots,q_n(t) \in L[t]$ irreducible polynomials ...
5
votes
4answers
316 views

What is the meaning of “algebraically indistinguishable”

I heard the term couple of times (in Field theory class and book), for example: The different roots of $p(x)=x^3-2$ are "algebraically indistinguishable". I understand the meaning intuitively, but ...
4
votes
3answers
396 views

Equivalence of Archimedian Fields Properties

I'm trying to prove that an Archimedian field is a subfield of the real numbers, my plan is to use the fact that the rationals are dense within the field and their Dedekind completion is the real ...
3
votes
2answers
139 views

Sum and Product of two transcendental numbers cannot be simultaneously algebraic

If $\alpha$ and $\beta$ are real number and $\alpha$ and $\beta$ are transcendental over $\mathbb Q$, show that $\alpha \beta$ or $\alpha +\beta$ is also transcendental over $\mathbb Q$ Attempt: ...
3
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0answers
140 views

Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly ...
3
votes
2answers
178 views

Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$?

Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$ for some $n$, where $\zeta_n=e^{2\pi i/n}$? I think the answer is no, but I can't give a full proof. Assume the contrary, we then have ...
3
votes
1answer
942 views

Construct a finite field of 16 elements and find a generator for its multiplicative group.

Construct a finite field of 16 elements and find a generator for its multiplicative group. Find all generators of multiplicative group. Very obvious Construction of a field with 16 elements according ...
3
votes
2answers
490 views

Purely inseparable extension

Let $F\subset K$ be an algebraic field extension. Is the set of all elements of $K$ that are purely inseparable over $F$ necessarily a subfield of $K$?
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3answers
68 views

Simple field extension inequality proof

Let $\alpha \in \mathbb{C}$ be algebraic over $\mathbb{Q}$ and $F\subseteq \mathbb{C}$ be a subfield. Prove that $[F(\alpha):F]\leqslant [\mathbb{Q}(\alpha):\mathbb{Q}]$. This looks like a problem ...
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vote
2answers
60 views

Let $\alpha \in \overline{\Bbb Q}$ a root of $X^3+X+1\in\Bbb Q[X]$. Calculate the minimum polynomial of $\alpha^{-1}$ en $\alpha -1$.

Let $\alpha \in \overline{\Bbb Q}$ a root of $X^3+X+1\in\Bbb Q[X]$. Calculate the minimum polynomial of $\alpha^{-1}$ en $\alpha -1$. I don't really understand how to get started here. I know ...
1
vote
2answers
106 views

In characteristic $p$, the field extension $k(X,Y)$ over $k(X^p,Y^p)$ has degree $p^2$

Let $k$ be a field with characteristic $p>0$, $L=k(X,Y)$ be the field of rational fractions of two variables over $k$. Let $K=k(X^p,Y^p)$. Prove that $[L:K]=p^2$ Help me a hint to prove it. ...