Exercise 1.2.1:
Verify that the set of complex numbers described in Example 4 is a subfield of
Solution: Let
is in is in- If
and are in then so is - If
is in then so is - If
and are in then so is - If
is in then so is
For 1, take
For 2, take
For 3, suppose
For 4, suppose
For 5, suppose
Exercise 1.2.2:
Let
Solution: Yes the two systems are equivalent. We show this by writing each equation of the first system in terms of the second, and conversely.
Exercise 1.2.3:
Test the following systems of equations as in Exercise 2.
and
Exercise 1.2.4:
Test the following systems as in Exercie 2.
Call the two equations in the first system
Exercise 1.2.5:
Let
- An operation is commutative if the table is symmetric across the diagonal that goes from the top left to the bottom right. This is true for the addition table so addition is commutative.
- There are eight cases. But if
or then it is obvious. So there are six non-trivial cases. If there’s exactly one and two ’s then both sides equal . If there are exactly two ’s and one then both sides equal . So addition is associative. - By inspection of the addition table, the element called
indeed acts like a zero, it has no effect when added to another element. so the additive inverse of is . And so the additive inverse of is . In other words and . So every element has an additive inverse.- As stated in 1, an operation is commutative if the table is symmetric across the diagonal that goes from the top left to the bottom right. This is true for the multiplication table so multiplication is commutative.
- As with addition, there are eight cases. If
then it is obvious. Otherwise at least one of , or must equal . In this case both and equal zero. Thus multiplication is associative. - By inspection of the multiplication table, the element called
indeed acts like a one, it has no effect when multiplied to another element. - There is only one non-zero element,
. And . So has a multiplicative inverse. In other words . - There are eight cases. If
then clearly both sides equal zero. That takes care of four cases. If all three then it is obvious. So we are down to three cases. If and then both sides are zero. So we’re down to the two cases where and one of or equals and the other equals . In this case both sides equal . So in all eight cases.
Exercise 1.2.6:
Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.
Solution: Write the two systems as follows:
Exercise 1.2.7:
Prove that each subfield of the field of complex numbers contains every rational number.
Solution: Every subfield of
Now
Exercise 1.2.8:
Prove that each field of characteristic zero contains a copy of the rational number field.
Solution: Call the additive and multiplicative identities of
Define
For
Call this map
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