# Questions tagged [field-theory]

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 topic. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

9,207 questions
Filter by
Sorted by
Tagged with
38 views

18 views

### In ZFC there exists a perfect field of given positive characteristic and given infinite cardinality

Fix a prime number $p$. In ZFC does there exist a perfect field of characteristic $p$ of any infinite cardinality? I know some constructions of fields of characteristic $0$ of arbitrary cardinality ...
31 views

### Why $X$ is a primitive r-th root of unity in lemma 4.7 Prime is in P? [duplicate]

Let $Q_r(X)$ be the $r^{th}$ cyclotomic polynomial over $F_p$. Polynomial $Q_r(X)$ divides $X^r-1$ and factors into irreducible factors of degree $o_r(p)$. Let $h(X)$ be one such irreducible factor. ...
20 views

### Question about finite extensions and splitting fields

Apparently any finite extension of a field with characteristic 0 is of the form $F(a)$ for some algebraic $a$. But wouldn't this mean that any two zeros of a polynomial (e. g. over $\mathbb{Q}$) would ...
56 views

### Show that $\mathbb{Q}(\sin\theta)$ is a field

QUESTION: Show that $F_{\theta}=\mathbb{Q}(\sin\theta); \theta\in\mathbb{R}$ is a field. Moreover, Show that $E_{\theta}=\mathbb{Q}(\sin\frac{\theta}{3}); \theta\in\mathbb{R}$ is a field extension ...
37 views

19 views

### Residue field and value group of an algebraically closed valued field [closed]

Why is the residue field of an algebraically closed valued field algebraically closed and the value group is divisible?
43 views

### Proof Explanation *Field and Galois Theory*

The following is with regards to Lemma 3.18 from Field and Galois Theory by Patrick Morandi. Let $\sigma : F \to F'$ be a field isomorphism, $K$ be a field extension of F, and $K'$ be a field ...
46 views

### Elements of $E^{\times},\cdot$ of the quotient ring $E:= \frac{\mathbb{Z}_3[X]}{\langle x^2 + x + 2\rangle}$

Consider the field $E:= \frac{\mathbb{Z}_3[X]}{\langle x^2 + x + 2\rangle}$. If I'm right the elements of the quotient ring can be found as: $$a_0 + a_1x + \langle x^2 + x + 2\rangle.$$ So we got the ...
43 views

### Galois group of order $n!$

Suppose $f\in\mathbb{Q}[x]$ has degree $n$, and let $K$ be the splitting field of $f$ over $\mathbb{Q}$. Suppose the Galois group of $K/\mathbb{Q}$ is $S_n$ with $n\geq 3$. It is not hard to show that ...
27 views

34 views

### In a finite field there exists an irreducible polynomial of degree at least $n$ for $n \in \mathbb{N}$

This question already has an answer here, but I'm looking for a solution that doesn't use field extensions. It's relatively easy to find a polynomial without zeros for every $n$ but as far as I know ...