Havlishkq

2022-02-15

A rational function f in n variables is a ratio of 2 polynomials,

$f({x}_{1},\dots {x}_{n})=\frac{p({x}_{1},\dots {x}_{n})}{q({x}_{1},\dots {x}_{n})}$

where q is not identically 0. The function is called symmetric if

$f({x}_{1},\dots {x}_{n})=f({x}_{\sigma \left(1\right)},\dots {x}_{\sigma \left(n\right)})$

for any permutation$\sigma$ of $\{1,\dots ,n\}$ .

Let F denote the field of rational functions and S denote the subfield of symmetric rational functions. Suppose the coefficients of polynomials are all real numbers.

Show that F=S(h), where$h={x}_{1}+2{x}^{2}+\dots +n{x}_{n}$ . In other words, show that h generates F as a field extension of S.

Attempt at Solution:

1.Can't seem to get very far with this one. I know that F is a finite extension of S of degree n! and the Galois group of the extension is$S}_{n$ .

2.Using h and the 1st symmetric function$x}_{1}={x}_{1}+{x}_{2}+\dots +{x}_{n$ , we see that $h-{s}_{1}={x}_{2}+2{x}^{3}+\dots (n-1){x}_{n}\u03f5S\left(h\right)$ .

3.Can't seem to find a good way to use the other symmetric functions$s}_{2},\dots ,{s}_{n$ .

where q is not identically 0. The function is called symmetric if

for any permutation

Let F denote the field of rational functions and S denote the subfield of symmetric rational functions. Suppose the coefficients of polynomials are all real numbers.

Show that F=S(h), where

Attempt at Solution:

1.Can't seem to get very far with this one. I know that F is a finite extension of S of degree n! and the Galois group of the extension is

2.Using h and the 1st symmetric function

3.Can't seem to find a good way to use the other symmetric functions

razlikaml42

Beginner2022-02-16Added 5 answers

According to Galois theory, since $S\subset S\left(h\right)\subset F,S\left(h\right)$ is $F}^{H$ , the field of elements of F fixed by some subgroup H of $S}^{H$ . Since h is only fixed by the identity automorphism, $H=\left\{id\right\}$ , and S(h)=FZSK.

Pregazzix2a

Beginner2022-02-17Added 9 answers

In more detail:

Let P be the minimal polynomial of h over S and let$\sigma$ be in $S}^{H$ , so that $\sigma \left(h\right)={x}_{{i}_{1}}+2{x}_{{i}_{1}}+\dots +n{x}_{{i}_{1}}$ . Since the coefficients of P are in S, $\sigma \left(P\right)=P$ , so $0=\sigma \left(0\right)=\sigma \left(P\left(h\right)\right)=\sigma \left(P\right)\left(\sigma \left(h\right)\right)=P\left(\sigma \left(h\right)\right)$ , thus $\sigma \left(h\right)$ is also a root of P.

Since all the$\sigma \left(h\right)$ are pairwise distinct, P has degree at least n!, thus the extension S(h) over S is at least of degree n!

But$S\left(h\right)\subset F$ , and F is also of degree n! over S, thus those two fields are equal.

Let P be the minimal polynomial of h over S and let

Since all the

But

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

Which operation could we perform in order to find the number of milliseconds in a year??

$60\cdot 60\cdot 24\cdot 7\cdot 365$ $1000\cdot 60\cdot 60\cdot 24\cdot 365$ $24\cdot 60\cdot 100\cdot 7\cdot 52$ $1000\cdot 60\cdot 24\cdot 7\cdot 52?$ Tell about the meaning of Sxx and Sxy in simple linear regression,, especially the meaning of those formulas

Is the number 7356 divisible by 12? Also find the remainder.

A) No

B) 0

C) Yes

D) 6What is a positive integer?

Determine the value of k if the remainder is 3 given $({x}^{3}+k{x}^{2}+x+5)\xf7(x+2)$

Is $41$ a prime number?

What is the square root of $98$?

Is the sum of two prime numbers is always even?

149600000000 is equal to

A)$1.496\times {10}^{11}$

B)$1.496\times {10}^{10}$

C)$1.496\times {10}^{12}$

D)$1.496\times {10}^{8}$Find the value of$\mathrm{log}1$ to the base $3$ ?

What is the square root of 3 divided by 2 .

write $\sqrt[5]{{\left(7x\right)}^{4}}$ as an equivalent expression using a fractional exponent.

simplify $\sqrt{125n}$

What is the square root of $\frac{144}{169}$