Mathematics, 31.07.2019 17:30 gonzalesnik

Prove that x 2 + y2 - 1 is irreducible in q[ x, y].

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Suppose that the polynomial x^2+y^2-1 is not irreducible.
It means that there exists two fractions such that :
$(\frac{p}{q})^2+( \frac{s}{t} )^2 -1=0\\\text{rearranging the above formula we get:}\\p^2t^2=q^2(t^2-s^2)\\\text{since q and p are primes each other, we deduce that }\\\text{q divide t meaning that we can write }t=a\times t\text{ for an integer a}\\\text{rearranging again the formula and using our finding we get:}\\q^2s^2=a^2q^2(q^2-p^2)\\\text{Same argument as above, we deduce that q divide s.}$
We found that q divide t and q divide s, but s and t are supposed to be prime each other. It is a contradiction.
The polynomial x^2+y^2-1 is then irreducible in q[x,y]

Answer from: Quest

correct answer:

b.(19,-2)

Answer from: Quest

Answer: question no. 1 -a. yes, the sides are in the ratio 2 : 5.question no. 2 -b. 59°question no. 3 -c. angles x and y are remote interior angles.question no. 4 -d. 90°, 45° and 45°question no. 5 -b. mn and qrstep-by-step explanation: question no. 1 -rs : xu = 3 : 7.5 = 2 : 5rt : x? = 6.4 : 16 = 2 : 5st : u? = 6 : 15 = 2 : 5as the ratios of all sides are equivalent, the triangles must be congruent and the sides must be proportional.therefore, the answer is: a. yes, the sides are in the ratio 2 : 5.question no. 2 -in the second diagram, angle b is the only letter not shown. therefore, the number must be equivalent to angle b.if we substitute the value of angle b as being 22° in the first diagram, we can now use the angle sum of a triangle to obtain the value of angle a.angle sum of a triangle = 180°therefore: let angle a = aa + 99 + 22 = 180a = 180 - 99 - 22a = 59°therefore, the answer is: b. 59°question no. 3 -angles x and y don't share the vertex of the triangle with the exterior angle w. remote interior angles don't share a vertex of a triangle with an exterior angle. this means that angles x and y are remote interior angles.therefore, the answer is: c. angles x and y are remote interior angles.question no. 4 -the sum of all interior angles of a triangle must be equivalent to 180°.90°, 45° and 45° are the only set of angle measures that are equivalent to 180°.therefore, the answer is: d. 90°, 45° and 45°question no. 5 -the only pair of sides stated in the options which are corresponding are mn and qr.therefore, the answer is: b. mn and qr.

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Prove that x 2 + y2 - 1 is irreducible in q[ x, y]....