Let $G$ be a finite cyclic group of order $p$. The tate cohomology groups $\hat{H}^\*(G, \mathbb{F}\_p)$ are defined using a complete resolution of $\mathbb{F}$ as $\mathbb{F}\_pG$-module.


There is a cup product on $\hat{H}^\*(G,\mathbb{F\_p})$, so I want to find the structure of this ring. 


In this MO question


[Examples of Tate cohomology rings](https://mathoverflow.net/questions/78287/examples-of-tate-cohomology-rings) 


it is stated the integer case; however, I have not been able to find a reference to that statement.


For the integer case, there is an isomorphism $\hat{H^n}(G,\mathbb{Z}) \cong \hat{H^{n+d}}(G,\mathbb{Z})$ given by the cup product with some element $u$ of degree $d$ (the period). **Does this also holds for the case $\mathbb{F}\_p$**?


Also, what techinque is used to compute the period of the tate cohomology and how to describe the ring?. For example, if $G$ is a cyclic group of order $2$, I think that $\hat{H}^\*(G, \mathbb{F}\_2) \cong \mathbb{F}\_2[t,t^{-1}]$ because degreewise, $\hat{H^n}(G,\mathbb{F}\_2) = \mathbb{F}\_2$.



Cup product in Tate Cohomology Ring
In Chapter XII.7 of the book 


*Cartan, Henri; Eilenberg, Samuel*, Homological algebra, Princeton Mathematical Series. 19. Princeton, New Jersey: Princeton University Press xv, 390 p. (1956). [ZBL0075.24305](https://zbmath.org/?q=an:0075.24305).


(starting on page 250) they calculate the cup products $\hat{H}^\*(G;A)\otimes \hat{H}^\*(G;A')\to \hat{H}^\*(G;A\otimes A')$ explicitly, where $A,A'$ are $\mathbb{Z}G$-modules, using a map of resolutions. This gives a reference for the integer case, and probably it's not too hard to adapt thier argument to get the calculation for $\mathbb{F}\_p$.


