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How could I work with polynomial fields?

I have the simple program:

a = var('a')
b = 1/a
for i in (1..5):
  c = b+1
  b = 1/c
  print b,",", c

The result is:

1/(1/a + 1) , 1/a + 1
1/(1/(1/a + 1) + 1) , 1/(1/a + 1) + 1
1/(1/(1/(1/a + 1) + 1) + 1) , 1/(1/(1/a + 1) + 1) + 1
1/(1/(1/(1/(1/a + 1) + 1) + 1) + 1) , 1/(1/(1/(1/a + 1) + 1) + 1) + 1
1/(1/(1/(1/(1/(1/a + 1) + 1) + 1) + 1) + 1) , 1/(1/(1/(1/(1/a + 1) + 1) + 1) + 1) + 1

Is there a way to get simpler expressions?