# Let F be a field and consider the ring of polynominals in two variables over F,F[x,y]. Prove that the functions sending a polyomial f(x,y) to its degree in x, its degree in y, and its total degree (i.e, the highest i+j where x^iy^i appears with a nonzero coefficient) all fail o be norm making F[x,y] a Euclidean domain.

Question
Abstract algebra
Let F be a field and consider the ring of polynominals in two variables over F,F[x,y]. Prove that the functions sending a polyomial f(x,y) to its degree in x, its degree in y, and its total degree (i.e, the highest i+j where $$\displaystyle{x}^{{i}}{y}^{{i}}$$ appears with a nonzero coefficient) all fail o be norm making F[x,y] a Euclidean domain.

2021-01-05
Let D be a domain. A non-negative integer valued function $$\displaystyle{N}:{D}-{\left\lbrace{0}\right\rbrace}$$ is Euclidean , if given a, b in D there exist $$\displaystyle{q}{\quad\text{and}\quad}{r}\in{D}$$ such that a=bq+r, with either r=0 or N(r) To show that f(x,y) going to highest degree in x is not a Euclidean function. (By symmetry) , the same argument shows that f(x,y) going to highest degree in y is not a Euclidean function.
$$\displaystyle{N}:{F}{\left[{x},{y}\right]}-{\left\lbrace{0}\right\rbrace}\rightarrow{\left\lbrace{0},{1},{2},{3},\ldots\right\rbrace}$$
N(f(x,y)) = highest degree wrt x
Claim: N is not Euclidean. Consider a=x+y, b=y. If N were Euclidean, $$\displaystyle\exists{q}{\left({x},{y}\right)},{r}{\left({x},{y}\right)}$$ with a=bq+r,r=0 or N(r) Now $$\displaystyle{a}={b}{q}+{r}\Rightarrow{x}+{y}={p}{\left({x},{y}\right)}{y}+{r}{\left({x},{y}\right)}$$
Comparing degrees, we deduce
p(x,y)=1 and r(x,y)=x.
But N(r)=1, whereas N(b)=0
so, N(r) So, N is not a Euclidean norm
Proving that the total degree function is also not a Euclidean norm
$$\displaystyle{N}:{F}{\left[{x},{y}\right]}-{\left\lbrace{0}\right\rbrace}\rightarrow{\left\lbrace{0},{1},{2},{3},\ldots\right\rbrace}$$
N(f(x,y)) = total degree
Claim: N is not Euclidean. Consider $$\displaystyle{a}={x}+{y}^{{2}},{b}={x}$$. If N were Euclidean, $$\displaystyle\exists{q}{\left({x},{y}\right)},{r}{\left({x},{y}\right)}$$ with a=bq+r,r=0 or N(r) Comparing degrees, we deduce
q(x,y)=1 and r(x,y)=y^2
But N(r)=2, whereas N(b)=2
So, N(r) So, N is not a Euclidean norm

### Relevant Questions

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Express your answer in terms of $$\displaystyle{C}_{{d}},{A},$$ and speed v.
Part B:
A certain car has an engine that provides a maximum power $$\displaystyle{P}_{{0}}$$. Suppose that the maximum speed of thee car, $$\displaystyle{v}_{{0}}$$, is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power $$\displaystyle{P}_{{1}}$$ is 10 percent greater than the original power ($$\displaystyle{P}_{{1}}={110}\%{P}_{{0}}$$).
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The top speed is limited by air drag.
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Express the percent increase in top speed numerically to two significant figures.
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When a gas is taken from a to c along the curved path in the figure (Figure 1) , the work done by the gas is W = -40 J and the heat added to the gas is Q = -140 J . Along path abc, the work done by the gas is W = -50 J . (That is, 50 J of work is done on the gas.)
I keep on missing Part D. The answer for part D is not -150,150,-155,108,105( was close but it said not quite check calculations)
Part A
What is Q for path abc?
Express your answer to two significant figures and include the appropriate units.
Part B
f Pc=1/2Pb, what is W for path cda?
Express your answer to two significant figures and include the appropriate units.
Part C
What is Q for path cda?
Express your answer to two significant figures and include the appropriate units.
Part D
What is Ua?Uc?
Express your answer to two significant figures and include the appropriate units.
Part E
If Ud?Uc=42J, what is Q for path da?
Express your answer to two significant figures and include the appropriate units.
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