# Viewing Polynomials from k[x,y] as Elements of k[y][x]

• NoDoubts
In summary, a polynomial f from k[x,y] with x appearing in positive degree can also be viewed as an element of k(y)[x], where the coefficients are rational functions of y. This is because a polynomial is also a rational function. The use of rational functions allows for a simple format of k(y)[x] as a ring of polynomials over a field. Additionally, if f is irreducible in k[x,y], it remains irreducible in k(y)[x] due to the ability to multiply by common denominators of the coefficients. This was proven by Gauss.
NoDoubts
"Let f is a polynomial from k[x,y], where k is a field. Suppose that x appears in f with positive degree. We view f as an element of k(y)[x], that is polynomial in x whose coefficients are rational functions of y."

I think I am missing something...why do we need rational functions here? can't we represent any polynomial from k[x,y] as an element of k[y][x] i.e. polynomial in x whose coefficients are polynomials of y?

I'm guessing here given I don't know from where you are getting this but...

Recall that a polynomial is also a rational function just as an integer is also a rational number.

I think the idea is to keep the format of k(y)[x] as a ring of polynomials (in x) over a field. So the y-polynomial coefficients are treated as elements of the larger ring of rational functions of y.

Yes, k[X,Y]=k[X][Y]=k[Y][X].

k[X,Y] is NOT equal to k(Y)[X].

yes, later on it says that "if f (polynomial from k[x,y]) is irreducible then it remains irreducible as an element of k(y)[x]"

can someone explain to me why it is so?

...I guess, if we assume f = gh where g,h are from k(y)[x] then we can multiply both sides by the product of common denominators of coefficients of g and h, So that we get f c(y) = g_1 h_1, where c(y), g_1, and h_1 are polynomials from k[x,y].

Now, how does this imply that f is not irreducible??

## 1. What does it mean to view polynomials from k[x,y] as elements of k[y][x]?

Viewing polynomials from k[x,y] as elements of k[y][x] means that we are representing polynomials with two variables, x and y, as elements of a polynomial ring in one variable, y, with coefficients from another polynomial ring in one variable, x. In other words, we can think of a polynomial in two variables as a polynomial in y with coefficients that are themselves polynomials in x.

## 2. Why would we want to view polynomials in this way?

Viewing polynomials from k[x,y] as elements of k[y][x] can make certain calculations and manipulations easier. For example, we can use techniques and algorithms from polynomial rings in one variable to simplify or factor polynomials in two variables.

## 3. Can we always view polynomials in this way?

No, we can only view polynomials in this way if the coefficients of the polynomial are from a field. This is because the polynomial ring in one variable, y, must be a field, and fields can only have polynomials with coefficients from the same field.

## 4. Is there a difference between viewing polynomials from k[x,y] as elements of k[y][x] and just considering them as polynomials in two variables?

Yes, there is a difference. When we view polynomials from k[x,y] as elements of k[y][x], we are considering them as elements of a specific algebraic structure with defined operations and properties. On the other hand, when we just consider them as polynomials in two variables, we are not necessarily focusing on their specific algebraic structure.

## 5. How does viewing polynomials in this way relate to multivariate polynomials?

Viewing polynomials from k[x,y] as elements of k[y][x] is a way of representing multivariate polynomials in a different form. It allows for easier manipulation and calculation of these polynomials, but ultimately they are still multivariate polynomials with two variables, x and y.

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