Show that 1 3x epsilon Z9x is a unit by finding a polynomia

Show that 1 + 3x epsilon Z_9[x] is a unit by finding a polynomial g epsilon Zg[x] such that (1 + 3x) middot g = 1. In Z_5[x]. the constant polynomials 1, 2, 3. and 4 are units. (Do you see why?) Does Z_5[x] contain any other units? Why or why not?

Solution

(a) The multiplicative inverse of 1+3x is 1-3x

(1+3x)(1-3x) = 1-9x² = 1-0x² = 1

Hence 1-3x = 1+6x is multiplicative inverse of 1+3x in Z₉[x] and so 1+3x is a unit in Z₉[x]

(b) 1 has multiplicative inverse 1 in Z₅[x]

2 has multiplicarive inverse 3 because 2.3 = 1 in Z₅[x]

3 has multiplicative inverse 2 because 3.2 = 6 =1 in Z₅[x]

4 has multiplicative inverse of 4 because 4.4 = 16 =1 in Z₅[x]

Now we know that Z_p[x] is a field for any prime p

Hence Z₅[x] is a field

A polynomial f(x) in field F[x] is a unit if and only if it is a non zero constant polynomial

And the non zero constant polynomials in Z₅[x] are only 1,2,3,4

Hence these are only units in Z₅[x]